The calculators you are seeing are all based on relatively straight forward formulas that you can find in many methods text books that cover survey research. All the that the formula requires is estimates of the population size and the variance of the item you will be measuring. Because most people don't have good numbers for the variance, the formulas typically assume the maximum variance that you would observe on a Yes-No question.

A good general source on surveys, which includes a chapter on sampling is:

Fowler, Survey Research Methods from SAGE.

It is not possible to answer this question in advance, other than the general recommendation of "get as many as you possibly can." However, I can ask a number of questions that might help you decide.

1) What is your study area? Is it the several million farmers in Africa, the hundred thousand or so in Ethiopia, or the several dozen in one river valley?

2) What is your return rate? If you give a farmer a survey, will you always get them to complete the entire survey?

3) What are your questions like? Are they all yes-no, Likert scales, or something else? Some questions are just more variable, and how you ask may increase or decrease that variability. So "do you raise goats" is less variable than "what is the primary animal you raise?"

The more variable your data, the larger the required sample size. 

It is also important that you sample enough of the population of interest. Sampling 2000 farmers from each country in Africa might be insufficient, yet sampling 39 of 40 farmers in the single valley of interest may be perfect. There are may other ways of restricting the sampling to make the problem more manageable. You have restricted the sample from all farmer to people raising animals, but maybe you are more interested in farmers with family incomes less than 8,000 Birr/year. Be aware that this works in both ways -- so if you sample a single valley, and no one in that valley makes more than 8,000 Birr/year, then you have inadvertently placed that restriction on your conclusions.

Note, that it is possible that the ideal sample size based on the expected variability in the response is larger than the total population of farmers that can be sampled.

Try searching the internet for research papers that poll farmers. Find ten or so articles and look at the sample size they used. Try and also determine the population size they targeted. Think of this as a lower bound, and try to do better. Keep in mind that what is published got past the editor and reviewers, that does not mean that it was also great science. We hope that the review process improves the science at least on a global scale, but some questionable stuff gets through anyway.

The aim of your survey is to identify the major health problems of  livestocks of farmers. For calculation of sample size, it is essential that you should have some rough idea about the list of health problems that are common among livestocks and their proportions. Ultimately, you may wish to list them    and may give the proportions suffering. In such a situation the  formula used could be S = 4pq/L^2 where P is the proportion of livestock supposed (a rough guess) to be suffering from a specified major health problem. q is equal to (1-p) and L is the accuracy with which you want to have estimate. Usually, it is expressed as % of P, can be 10%, 15% or 20% as the investigator wants it. For example I wish to assess the smoking problem among urban population. A rough estimates of it could be 30%. This means P =0.3 and q=0.7 and L you have taken it to be 20% means you want an accurancy of plus minus 20% in your p estimate. So the sample size could be then S = 4*0.7*0.3/(0.7*0.2)^2 = 233. Therefore, Your first job will be to get a rough idea about the proportion of livestock suffering from the specified health problem. In very general terms, I want to estimate all the health problems that are 5% or more among the livestocks. This means my p is 0.05, q =0.95. Now using the formula calculate the sample size which comes out to be 1900. I hope this answer your question.

Excellent. This shows that the answer given is based on the assumptions made. If you are interested in a point estimate (say disease prevalence in a 100 km2 area in one valley) then Ramnath's suggestion is great. However, if you are sampling a larger area, then you will need to do this over and over again. In Florida (USA), the number of psyllids carrying a citrus disease ranged from 60% to less than 0.5% in a space of 50 miles. I need a very different sample size at each location. I don't know how prevalence changes within the geographic scale of this project, so this may or may not be a problem.

In my previous answer I assumed that the survey was more about how farmers view the disease problem, or how they try and manage it. Possibly how farm practices/attitudes change in different parts of Ethiopia and how effective different practices are at managing disease problems.

Of course this could be a study on how accurate farmers are at estimating their disease issues. So part 1 is a survey, and part two is taking blood samples and using molecular methods to test for disease. One then looks at the correlation between then two values and how other factors influence farmer accuracy.

The calculators you are seeing are all based on relatively straight forward formulas that you can find in many methods text books that cover survey research. All the that the formula requires is estimates of the population size and the variance of the item you will be measuring. Because most people don't have good numbers for the variance, the formulas typically assume the maximum variance that you would observe on a Yes-No question.

A good general source on surveys, which includes a chapter on sampling is:

Fowler, Survey Research Methods from SAGE.

Thank you all for your answers