Sorry I missed your reply here, but on the off-change you get this: a random sample of 10 people is unbiased for the entire population. A random sample of 1 person is unbiased for the entire population. The limitation of very low sample size is that it is going to produce high variance; on average it will be correct, but across multiple surveys our estimates will move around a lot. We can define a margin of error around our estimate to formally account for this. If what we're interested in is a binary yes/no answer, using the classical calculations and without trying to add more complexity for you, the margin of error is +/- 1/sqrt(n), so a sample of 100 people gets you in a qualitative ballpark of whether a property is very common, common, rare, or very rare. If you want to tell whether 69% and 71% reflect statistically different underlying propensities, you will need a much bigger sample. But if you just need to know "does there exist a decent swath of people who say yes to this?", a very small sample will do on average.
The full population size is generally irrelevant statistically (it applies to a "finite population correction"). Almost all the statistics you see presented publicly ignore FPCs by assuming the population is infinitely larger than the sample. Even when people take census reads of a full population they typically assume the population is a realized sample of a larger meta-population.
In general if you are seeing publicly presented poll data, sample size is not the thing that should be sending up red flags. Differential non-response and selection bias; sample frame matching or not matching the population; underlying noise in the conceptual measure; design effects caused by weighting; and many other components are all part of "TSE" (total survey error), which dwarfs the impact of sampling variation as a concern.
Now, mind you, subgroup analyses often do cut samples too finely... e.g., an N=500 sample subset to Men 65+ and people trying to make inferences but not properly reflecting either the sampling considerations or reduced subgroup sample size).
Let it also be noted from those classical calculations that larger samples are diminishing returns: N = 1000 -> 3.1% -> N = 5000 -> 1.4%. Five times the cost of surveying if not more, only a halving of the margin of error, and limited extra power to answer most real questions of interest.