It feels really uncomfortable that, if you have infinitely many people wearing red or blue hats that they can't see, then they can all guess their own hat color with only finitely many of them being wrong. Not to mention Banach-Tarski and a hundred other strange phenomena. This all militates toward rejecting AC.

But then, if we reject AC, we can have infinite sets that are incomparable (i.e., they're not isomorphic and neither is larger than the other). So pick your poison!

Thinking about such things for too long makes me feel grateful that I spend most of my time in the finite world (or, let's say, the world of continuous parameters that we only ever measure to finite precision, so that the statements we care about can ultimately be phrased arithmetically). In this world, because of e.g. the Shoenfield absoluteness theorem, anything that can be proved with the help of AC can also be proved without it.