There are frequentistic and Bayesian approaches to find "evidence" for the null hypothesis. But all need that you define not only a point null hypothesis (as it is usual in typical frequentist approaches, e.g. correlation = 0 or mean difference = 0), but a region of values, which you would consider a null effect, i.e. these values are so small that they are of no practical interest. For this you might consult some papers concerning the term Smallest Effect Size Of Interest (SESOI).

If you have defined such a region you might use one of these approaches:

1) The frequentistic Two One Sided Tests (TOST), see publications by Daniël Lakens for example.

2) The Bayesian Region Of Practical Equivalence (ROPE), see publications by John Kruschke for example

3) The Bayesian Bayes Factor (BF), see publications by Eric-Jan Wagenmakers for example.

All approaches have their pros and cons in my opinion and it depends what you are specifically looking for. I personally favour the ROPE approach, but this is only my opinion, since it follows the model workflow I would use. For a comparison of approaches see:

Linde, M., Tendeiro, J. N., Selker, R., Wagenmakers, E. J., & van Ravenzwaaij, D. (2021). Decisions about equivalence: A comparison of TOST, HDI-ROPE, and the Bayes factor. Psychological Methods, 28(3), 740.

and for a critique about their general finding and conclusion in favour of the BF:

Campbell, H., & Gustafson, P. (2021). re: Linde et al.(2021): The Bayes factor, HDI-ROPE and frequentist equivalence tests can all be reverse engineered--almost exactly--from one another. arXiv preprint arXiv:2104.07834.

There is virtually always an affect. Testing the null hypothesis is almost all a trivial effort where the result is known on first principles. Less or more? variable over time? of consequence? - these are not always obvious or known beforehand ... but zero affect? Not really an issue. Of course the affect is not zero when different populations are involved.

As Kim Iles points out, an effect of exactly zero is unlikely, a better test may be to test of whether the effect is less than what you think is a meaningful value. In bioequivalence, the blood levels of a drug from a standard product versus another, possibly generic, product are compared to test whether they differ by a specified amount, say 20%. (It doesn’t have to be zero.) The test is called the two one-sided tests procedure (TOST) (Schuirmann, Donald J. (1987-12-01). "A comparison of the Two One-Sided Tests Procedure and the Power Approach for assessing the equivalence of average bioavailability". Journal of Pharmacokinetics and Biopharmaceutics. 15 (6): 657–680.) There may be a version that applies when comparing a population mean to a fixed value, for you 0. There is an R function by Detlew Labes for the power of TOST (https://cran.r-project.org/web/packages/PowerTOST/PowerTOST.pdf) based on my paper “Power of two-one-sided tests procedure in bioequivalence”, May 1990, Journal of Pharmacokinetics and Biopharmaceutics. Perhaps there is version of TOST and its power for the case of a fixed comparator. It’s possible that Labes’ package has such a function.

Let me offer a little bit of a different perspective.

• Coming from an agricultural statistics background, it's common for an experiment to use a control treatment, and what I might call a "check" treatment --- though I suspect that there's a different term for this. For example, if you were testing different products to, say, get rid of weeds from a turfgrass plot, the three treatments you might be testing would be three herbicides. A control treatment might be applying no treatment (presumably to plots that are likely to get a good infestation of weeds, and the "check" treatment might be an standard treatment or an ideal treatment --- maybe in this case, weeding by hand where you remove essentially all of the weeds.
• This gives a range in the effects of treatments. That is, if you can see differences between the control and the herbicide treatments, you know your experiment has enough power to detect these effects.
• If you couldn't, for example, distinguish the treatments from the control or the "check" --- the ideal or standard treatment --- your experiment lacks power or is poorly designed.
• Plotting the data or means with error bars makes it clear to readers the relative merits of the treatments relative to the control and the "check".
• Also, in agricultural experiments the units are often an understandable quantity. That is, if the effect size is 0.1 tonnes / ha of grain corn, readers are likely to know what that means. So, without explicitly saying, "We consider an effect size of < 0.1 tonnes / ha to be practically unimportant", the reader has a sense of what the treatment differences actually mean.
• Also, in agricultural experiments, the costs of the treatments are somewhat understood by the readers. An old, established chemical treatment maybe relatively cheap. A new chemical treatment may be statistically better, but if it's more expensive, and the effect size is relatively small, the reader has a sense of whether this is "worth it" or not.

Just a bit of a different way of thinking about the question. And also, why I suspect, I never learned about standardized effect size statistics or equivalency testing in this sub-domain. I just think historically, these considerations were baked into doing experiments in a realm that the readers understood the practical importance of relative effects and treatment costs.

I would take issue with just one thing in Sal Mangiafico 's thoughtful reply: plotting means and error bars is inefficient, because you are using a lot of ink to display just two summary statistics. But worse, there is no sense of the amount of data, its dispersion or distribution shape. And error bars cause people to draw conclusions which may or may not be correct.

Excellent preprint here

Preprint What’s really wrong with bar graphs of mean values: variable...

And also see

Vail A, Wilkinson J. Bang goes the detonator plot! Reproduction. 2020;159(2):E3–4.

A boxplot superimposed or alongside the data gives a lot more information. Raincloud plots are even more information-rich, and very useful for times when the shape of the distribution is noteworthy, or differs between the groups being compared.

I'll add a couple of articles I like to the list Rainer Duesing gave you.

Walker E, Nowacki AS. Understanding equivalence and noninferiority testing. J Gen Intern Med. 2011 Feb;26(2):192-6. doi: 10.1007/s11606-010-1513-8. Epub 2010 Sep 21. PMID: 20857339; PMCID: PMC3019319.

https://pubmed.ncbi.nlm.nih.gov/20857339/

Shafiq N, Malhotra S. Superiority trials: raising the bar of null hypothesis statistical testing. Evid Based Med. 2015 Oct;20(5):154-5. doi: 10.1136/ebmed-2015-110280. Epub 2015 Sep 3. PMID: 26337934.

Article Superiority trials: Raising the bar of null hypothesis stati...

I hope this helps.

As a addition to my first post, a very recent paper about the problems of the HDI + ROPE approach by Kruschke and how to remedy it.

Etz, A., Chávez, A., Baroja, J. L., Medriano, K., & Vandekerckhove, J. (2023). The HDI+ ROPE decision rule is logically incoherent but we can fix it.

*strangely, I cannot reach the OSF website to give the link....

Rainer Duesing : here's the link https://files.osf.io/v1/resources/5p2qt/providers/osfstorage/65713e5402fa9631407d285b?action=download&direct&version=1