A bolt action is a little bit quiter than a semi-auto and so is a pump action .22. For quiet and legal, it is hard to beat .22 shorts in a bolt action rifle. You could also use BB caps or CB caps but they don't have very much power if you are wanting to hunt with them. I used to hunt squirrels with a .22 rifle and .22 short hollow points. You could sit under a tree and kill 2 or 3 from one tree. they didn't seem to be bothered by the little pop of the short and would be out very soon after you had just shot one out of the same tree.

Of course, you could put a noise supressor (silencer) on a bolt action rifle and shoot sub-sonic rounds but you had better be sure that you are in compliance with all of the laws concerning having and using a silencer. I'm pretty sure that they are not legal for hunting anywhere, though. So, you'd be limited to shooting targets or plinking at cans with it.