How to patch a hole in stained wood (oak) so that the patch material matches the wood as close as possible?
Using a plug is your best option.  Getting an exact match is pretty much impossible.  A drill press is the tool of choice for a perfect plug if is conjunction with a good hole saw bit or plug cutting bit.  If you can cut the plug from the same stained material in an out of sight location, this would save a lot of time.  Color matching is tough to do. Keep in mind that nobody will be looking for the hole after you patch it like you do.  You would be surprised how little people notice unless you point it out :)

Good luck.