In the absence of sample frame, the researcher cannot use direct method of sampling, but has to adopt indirect way/method of selecting sample size systematically from total population after giving proper justification for the size selected..
From a quick look at your profile, I will assume you are considering continuous quantitative data. Similar arguments may apply in an analogous way to other data, but the notes below assume continuous data.
If you do not have a frame of any sort, list or area, but can find data randomly, preferably within stratified random subpopulations based on characteristics which define each such stratum/group, ... and/or your data are associated with known auxiliary data for your population, ... then you can estimate means, but not totals, and you can estimate variances, if your unknown population size is great enough that you do not need to consider finite population correction (fpc) factors, and if you are reasonably certain that there are no unknown categories of the population of interest.
Sample size needs depend upon population standard deviations and sample designs. Procedures generally assume bias is negligible, though that is often likely a dubious assumption. If you look in a standard survey sampling book such as Cochran (noted below), you may find useful information by ignoring (or removing) fpcs, and assuming an infinite population.
Cochran, W.G(1977), Sampling Techniques, 3rd ed., John Wiley & Sons.
Blair, E. and Blair, J(2015), Applied Survey Sampling, Sage Publications.
And this one is analytical, yet surprisingly clearly written:
Lohr, S.L(2010), Sampling: Design and Analysis, 2nd ed., Brooks/Cole.
But basically, sample size needs are estimated by the size needed with relevant standard deviations to reach a desirable standard error or less. One way to estimate standard deviation by group (subpopulation/stratum, or population as a whole if appropriate) might be a pilot study.
Best wishes - Jim
PS - The following paper, written by Ken Brewer upon receiving the 2013 Waksberg Award, may be of interest. (Note that the one completely model-based example is for the classical ratio estimator, CRE, but there are other levels of heteroscedasticity that he notes in his book, and elsewhere, that could also be considered.)
Ken Brewer's Waksberg article:
Brewer, K.R.W. (2014), “Three controversies in the history of survey sampling,” Survey Methodology,
(December 2013/January 2014), Vol 39, No 2, pp. 249-262. Statistics Canada, Catalogue No. 12-001-X.
He believed in using probability sampling and models together, but as I've told others, Ken explains the different approaches. Ken's paper for the Waksberg Award is rather entertaining.
...
PSS -
Two more notes:
1) I see that one topic you have listed is "acceptance sampling," which is a related area regarding quality control/reliability, but I did not mention it above because I don't think that pertains to your question. I could be wrong.
2) Also, if your problem entails regression, then the population of interest in sample size estimation is the set of estimated residuals, or more appropriately, since it covers weighted least squares regression, the population of the estimated random factors of your estimated residuals. If normality is hoped for, that is with regard for residuals, not the y-values, which are often highly skewed.
Determining the sample sizes involve resource and statistical issues. Usually, researchers regard 100 participants as the minimum sample size when the population is large. However, In most studies the sample size is determined effectively by two factors: (1) the nature of data analysis proposed and (2) estimated response rate.
For example, if you plan to use a linear regression a sample size of 50+ 8K is required, where K is the number of predictors. Some researchers believes it is desirable to have at least 10 respondents for each item being tested in a factor analysis, Further, up to 300 responses is not unusual for Likert scale development according to other researchers.
Another method of calculating the required sample size is using the Power and Sample size program (www.power-analysis.com).