I wonder how much of this is just a function of what math you've ended up working with a lot.

Humans have really bad intuition about math. This shouldn't be that surprising. We evolved in a context where selection pressure was on finding mates and not getting eaten by large cats.

Speaking from personal experience as a mathematician (ok a grad student but close enough for this purpose) it isn't that uncommon for when I encounter a new construction that has some counterintuitive property to look at it and go "huh? Really?" and not feel like it works. But after working with the object for a while it becomes more concrete and more reasonable. This is because I've internalized the experience and changed my intuition accordingly.

There are a lot of very basic facts that don't involve infinite sets that are just incredibly weird. One of my favorite examples are non-transitive dice. We define a "die" to be a finite list of real numbers. To role a dice we pick a die a random number from the list, giving each option equal probability. This is a pretty good representation of what we mean by a dice in an intuitive set. Now, we say a die A beats a die B if more than half the time die A rolls a higher number than die B. Theorem: There exist three 6-sided dice A, B and C with positive integer sides such that A beats B, B beats C and C beats A. Constructing a set of these is a fun exercise. If this claim seems reasonable to you at first hearing then you either have a really good intuition for probability or you have terrible hindsight bias. This is an extremely finite, weird statement.

And I can give even weirder examples including an even more counterintuitive analog involving coin flips.

I just don't see "my intuition isn't happy with this result" to be a good argument against a theorem. All the axioms of ZF seem reasonable and I can get the existence of uncomputable ordinals from much weaker systems. So if there's a non-intuitive aspect here, that's a reason to update my intuition not to reduce my confidence in set theory.

ETA: If you want to learn more about this (and see solution sets for the three dice problem) see this shamelessly self-promoting link to my own blog or this more detailed and better written Wikipedia article.