Following Eric Wofsey's point, you probably are interested in things that don't involve the pathologies of arbitrary uncountable objects. Some people have argued that the only things "ordinary mathematicians" care about are things that are representable in second-order arithmetic. (This gives you arbitrary real numbers, arbitrary complete separable metric spaces, Borel and analytic sets on all of those, and similarly interesting amounts of algebraic stuff.) The project of Steven Simpson's book Subsystems of Second-Order Arithmetic is to analyze just what axioms are needed to prove which results. All of the things considered there are far weaker than ZFC. But it's interesting to discover that beyond the axioms of Peano Arithmetic and the existence of arbitrary recursive sets of natural numbers, there are exactly five natural strengths. That is, there are five levels such that very large numbers of theorems from ordinary mathematics fall at exactly one of the levels, where any theorem at one level can be proved by assuming any theorem at that level or higher. Interestingly, things like the Heine-Borel theorem and the Bolzano-Weierstrass theorem, which are often thought of as equivalent, actually fall at different levels.
Not everything falls exactly at these levels though. Some things do still depend on a version of the axiom of choice, which is above any of these five levels, and there are other results like Goodstein's theorem and Borel determinacy that are higher still (I believe).