Your original intuition, that only a countable subset of real numbers can be described or used in any way, is correct. The rest are just “there.” They exist, but we can’t really use them for anything.

It gets weirder. What is a set? For finite sets, we know it intuitively. But consider the Axiom of Choice. There is a consistent mathematics in which a choice set is a set, and one in which the same meta-mathematical object is not a set. (Unless, of course, ZF is inconsistent.)