There are lots of approached these days being followed , where there is minimum role of chemicals ,  though the world is divided , which one is the best , since food demand is on the continuous increasing side. some of the approaches like organic farming , biodynamic farming , natural farming , conservation agriculture ( allows limited use of chemicals ), integrated crop management , however , combination organic farming and conservation agriculture has produced much better dividends . This is such a topic , you can debate and debate . 

Following complete package of organic farming practice production and quality can be achieved. For example: Biodynamic farming practices can be adopted to answer the quistion..

You need to apply multiple inputs to meet the nutrient demand of any crop without using chemical fertilizers. You can make it 100%  successful.

 This may be achieved by utilizing the native fertility of soil, i.e by making the nutrients available / creating a conducive environment for the release of the nutrients for plant uptake by many means. However, a word of caution is that in log run it will cause mining of nutrients from soil, Hence there should be addition of inputs either it may be organic or inorganic. I agree with point told by Dr.AK Srivastava, that it should be a combination a both.Think about it and wisely use the soil for achieving maximum production with good quality

As we know attempting to separate organic & inorganic nutrients is difficult & of limited value because nature’s processes are continually cycling them from one form to another. All nutrients go through natural cycles, following various pathways to their final destination of being absorbed and utilized by plants that grow all the food for humans and animals. In the process, some, such as N, P, and S, cycle back and forth between the inorganic and organic pools. Therefore integrated nutrient management involving both organic and inorganic will definitely improve the quality and quantity of produce. Alone organic will help only for improvement of quality not quantity.

Crop rotation, improve organic matter, optimize use of water, specific species and cultivar 

There are lots of approaches to improve the quality and quantity of crops without the use of any chemicals and inorganic fertilizers. However, there is no one best method but rather a combination of different practices is required, in retrospect to a particular region.  Things like abiotic and biotic factors (soil type, climate) topography, crop type, etc should be considered.

This is to say that even though a set of practices works well in one area it may not be appropriate for another.

Generally, as it regards to soil nutrients the incorporation of organic matter is a good practice. However, the soil type, nutrient present, and crop requirement must be considered. As the heterogeneous nature of these organic materials vary in nutrient composition and decomposing rates and should be considered.   The use of plant growth promoting organisms such as mycorrhiza, P solubilizing and N fixing bacteria aids in fertility as well.

Intercropping with complimentary sharing plant such as N fixing incorporates nutrients and can help to reduce weeds (cover crops).

Biocontrol inoculum and crop rotation are some examples used to reduce plant pathogens and soil-borne diseases.

Even irrigation practices, for example, drip irrigation improves water use efficiency and help reduce the leaching of nutrients, particularly mobile macro nutrient like N.  Practices like these that directs water at the root system also helps to reduce weeds and harmful fungal growth, thus reducing the need for chemical control.

When depending on natural inputs instead of synthetic chemicals and pesticides it is imperative to improve the soil. This starts with a complete soil analysis. Continues with the development of a rotational plan. The analysis should include work on drainage. The way to effectively increase soil organic matter includes the use of perennial forage which can increase soil nitrogen and carbon. The liming of acid soils would be encourage and the use of rock minerals such as phosphates according to production goals. Animal and plant production can be mutually supportive and manures should be returned to the large as compost. Water features and windbreaks can be very important to sustainable farm. 

I think , ways and means to trigger nutrient cycling in a given agroecosystem is pivotal to experience expected results on soil health related issues and production sustainability over time and scale.

A lot of detail given above and in my opinion by using GMO's technology we can achieve the said targets !!

 The Rodale Institute Farming Systems Trial directly compares a conventional large scale farming approach with dependence on synthetic inputs compared to an approach where biological inputs replace the synthetic inputs of fertilizers and pesticides. When the farming system improves the soil by rotation yields in the two approaches are not different in normal precipitation years. In dry years the improved soil under biological approaches and focus on soil show improved performance under the drought stress. GMO technology has been designed around the use of synthetic inputs. GMO technology can do lots of things such as promote resistance to insect and diseases. What it cannot do is become a substitute for long term focus which will improve soil quality and health. In fact, the monoculture approach of GMO based systems eliminates the biological diversity with which a biological system requires for both high productivity and stability. It is debatable if the GMO focus has reduced the use of synthetic inputs as promoters suggested it would do. In fact, after decades of its use the statistics on the use of pesticides and fertilizers show little decrease in this dangerous dependency.  

There is a farming system called Agroforestry where woody perennials are deliberately grown on the same land management unit as crops and /or animals in a form temporal sequence or spatial arrangement.  The choice of the woody perennial depends on the farmers objectives and largely on the nutrient deficiencies of the soil. Example Leucaena  leococephala is a nitrogen fixing plant which when cut back onto the land would decompose and address the nitrogen deficiency of a soil. Different species addresses different problems of the soil and sustainable management of the system improves productivity and livelihood of farmers as other products  from the woody perennial serves other purposes. For instance leaves could also be used to feed livestock, poles for diverse uses could be derived from the trees and soil conservation is always achieved. It is very effective. Farmers who practiced this farming system had a tremendous improvement in their livelihood.

What we know is that whatever you harvest or take away from the field or farm is a mining of nutrients (macro & micro). So the higher the crop yield, the higher the nutrient mining, If a nutrient balance is not managed well, then you will have deficiencies in the plant and resultant poor growth, ultimate degradation of the soil. There are many ways to handle this, like improving nutrient efficiency of the plant (biotechnology, breeding...etc), the nutrients mined should be replaced in one way or another for the soil's capacity to support plant growth. Recycling of nutrients works well in a natural forest where nothing is removed, decomposition process releases the nutrients back to the soil, so under conventional systems you try replace the mined nutrients through fertilizer application, especially if your crop is high-yielding as the soil may not supply quantities of nutrients needed by the crop. Can one grow a crop with limited nutrients, yes it is possible, this now the role of research... 

High yielding bred disease resistant varieties of leguminous crops inoculated with nitrogen fixing bacteria (Rhizobium) + Phosphorous solubilizing bacteria (PSB) + PGPRs can be used in crop rotation with any other crop to maintain the soil health, nutrients, quality and quantity of the crops. Additionally, the crop residue can be used to incorporated in the soil and/or as a mulch for multi-directional benefits.

use biofertilizer or organic-fertilizer combine with land management 

Planting improved and high yielding varieties , regular care,  crop rotation along with green manure crops

Some crops are compatible with multi-cropping .In such cases; combination can improve the quality in many ways. Legumes and other crops cultivation is such practice. Inter cropping in between seasons also give the same results. This can help in many ways ex: pests control, fertilizer efficiency, avoid soil erosion, weed control etc. In the same way, the suitable mixing of livestock in to the system also helps to improve the crop yield. As an example Perennials like coconut can cultivate with pineapples while cattle keep in the same area. This will help to control weed and add fertilizer. The land extent, climate and agro ecological zones are needed to concern in such mixing systems. Another way is, improve the micro climate via irrigation practices ex: drip irrigation or sprinklers. This can help to improve the quality. As an example fruit cracking and blossom end rot of tomato can be controlled in that way. The shading or mulching for suitable crops may help to improve the quality .as an example cauliflower and cabbage head formation are sensitive to heat thus mulching is important. The moisture control may help to avoid fungi in many crops. Therefore, it is needed to control sunlight using proper shading methods. Nevertheless, control wind speed by using shelter tree belts can avoid wind damages. As an example in tea cultivation, it is frequently used such tree barriers to prevent damages to the leaves (especially in hilly areas). Proper harvesting methods are also important as an example in pepper cultivation; it is required to keep the buds and branches undamaged for the fruiting of next season. Pruning of damaged or old branches are important in some crops like tea, grapes. That will help to improve the harvest. Agronomic practices like above can use to improve both quality and quantity. It is always good to identify the crop and agronomy of crop before the practice because certain crops require different practices as an example dragon fruit requires a different facilitation for pollination and propagation.  

The way to improve the quality and quantity of crops without use of chemicals and fertilizers are as under;

increase input use efficiency

use of biofertilizers

use of bio pesticides

crop rotation

use of high yielding improved varieties

IPM, IDM,INM

use of organic manures etc.

I am supporting the answer of Mr Ardeshna N. J.  because he summarized the above researcher's ideas about this question. 

 Improving soil without necessarily using fertilizer can be achieved adopting any of the following;

1. Promote and encourage inter-cropping with leguminous fixing crops

2. Practice shift/rotation cropping

3. Compost making by way of converting community refuse as source of manure

4. Introducing insects management/survellance against pest

5. Planting disease resistant varities

6. Proper weed control to lower incidence of pest (manual control)

The quality and quantity of crops without use of chemical and fertilizers can be increased by adopting following measures;

1. Enrichment of soil and replenishment of depleted soil nutrient with application of biofertilizers, green manuring, organic manure, farm yard manure, burring of plant residue in soil etc.

2. Promoting adoption of inter-cropping with legume based crops

2. Crop rotation

3. Reduce depletion of soil nutrient by adopting an appropriate weed management strategy which include zero tillage, manual weed control, chemical weed control, cover crop etc.

4. Crop rotation with leguminous crops

5. Encourage farmers to adopt insect-pest and disease resistant varieties

6. Management of insects pests and diseases by adopting integrated pest management and integrated disease management practices which involve more use of bio pesticides with due weightage on biological control of pests and diseases

7. Management of irrigation by using modern methods of irrigation like drip and sprinkler method

8. Use of mulching

9. Reducing post harvest losses by adopting improved method of harvesting and timely harvesting to maintain quality of produce. It also include reduction of storage losses and deterioration of quality during storage by adopting scientific method of storage and management practices for control of pest during storage.

I think that another important point that has not been considered yet is about genetics. Rather than hybrid varieties, or GMOs a system that is not using chemical inputs would be better of with landrace varieties. These may not have the improved traits that have been selected for hybrid, or GMOs, yet they may be most adaptable to a specific agronomic context because in this environment is where they had the opportunity to adapt and evolve.

Please have a look at these useful links.

https://www.nytimes.com/1987/08/23/us/farming-without-chemicals-age-old-technologies-becoming-state-of-art.html