If you do give up the axiom of choice, is there a lot of cool/useful math that you'd also have to give up?
While the Banach-Tarski violates your intuitions, it is probably because it is more about how there are "multiple copies" of the real numbers in the real numbers, than anything resembling a real cut.
However, to me the "basic principle" is a similar argument. The important thing (for me) was to understand this is an argument about slicing up infinities in an "Infinite comb" type way, rather than anything resembling a "cut".
Reverse mathematics (http://en.wikipedia.org/wiki/Reverse_mathematics) is a reasonably recent branch of mathematics aiming to investigate those kinds of questions.
Simplest example: you can have a product of non-empty sets be empty.
* Let X and Y be sets. Then either they have the same cardinality, or one is smaller than the other.
* Let X be an infinite set. Then there is a bijection between X and the cartesian product of X with itself, X × X.
* Tychonoff's theorem: every product of compact topological spaces is compact.
As far as Tychonoff's theorem goes, you might find this paper interesting:
That said, it is a theorem that absolutely no theorem of number theory can possibly depend on the axiom of choice. So absolutely no "real" consequence can come from accepting or not the axiom of choice.