First, the question of "provability" is whether something can be proved using a given collection of axioms and rules.
That is, we are not concerned about whether a smart mathematician with great communication skills could convince us about the veracity of a statement. Instead, we are concerned about whether we can construct a sequence of statements culminating with the given statement, such that each step can be verified by a rather simple computer program, by matching it to the few predetermined patterns.
So it is perfectly okay to have a statement that is obviously true, but still cannot be proved using some set of axioms and rules. Think about it as a weakness of the given set of axioms and rules, rather than a property of the statement itself. In other words, we never talk about "provability of X" (except as a shortcut), but always about "provability of X in system S". Provability is a relation between X and S.
Therefore, Gödel's Theorem is not about statements that are unprovable per se (i.e. some more complicated version of "math will never be able to prove that 2 + 2 equals 4"), but rather it means "if you give me a non-contradictory system S of axioms and rules, I can create a statement X such that its veracity will be beyond the reach of S". Plus there is an algorithm for creating a specific X for given S. The generated X will be tailored to the given S. (For different systems S1, S2, S3 you would get different X1, X2, X3; and it's always S1 having a problem with X1, S2 with X2, and S3 with X3; but maybe X1 and X2 are easy to prove or disprove in S3.)
More specifically, the statement X generated by Gödel for system S is cleverly designed to be equivalent to "X cannot be proved by S" without being self-referential. To discuss the clever construction is beyond the scope of this comment.
Second, once we start talking about things like sets, we are now far outside the realm of reality. It doesn't make sense to ask whether "sets really have a property P", if the fact is that "in the first place, set's don't really exist".
It would be like trying to use experimental physics to verify whether fairies are heavier than 10 grams. You simply can't, because there are no fairies to measure. So if you have two competing theories of physics-including-supernatural, one saying that fairies are heavier than 10 grams, and the other saying they are not, well, you have a problem that cannot be resolved experimentally. A set theory with the Axiom of Choice, and a set theory with some incompatible axiom, are two such theories: they agree about everything that really exists, and they disagree about the properties of fairies... ahem, infinite sets. You can't resolve this conflict by talking about what really exists; and the arguments about what doesn't exist are by their nature arbitrary. (You can't prove internal inconsistency, because both theories are internally consistent. They just disagree with each other.)
So how can we study sets if they are not real? By defining axioms and rules, and examining which statements can be proved using given axioms and rules. "ZF" is such a set of axioms; there are statements you can prove using it, there are statements you can disprove, and there are statements where the system provides no answer.
The underlying reason is that if you imagine a Platonic realm where all abstractions allegedly exist, the problem is that there are actually multiple abstractions ["models"] compatible with ZF, but different from each other in many important ways. In some of them, Axiom of Choice is true. In others, it is false. This is what it means that Axiom of Choice can be neither proved nor disproved in ZF. The problem is that "sets" have never been unambiguously defined in the first place!
However, adopting the Axiom of Choice (or any of its competitors) won't actually solve the problem. There will still remain many abstractions compatible with ZFC (or whatever), but different from each other. So you will sooner or later find other exciting properties of the fairies... ahem, infinite sets, that can be neither proved nor disproved by ZFC (or whatever). The problem, again, is that we still don't have an unambiguous definition of "sets".
And... but I am way out of my depth here... maybe we can't have an unambiguous definition of "sets", ever, because of the Gödel Theorem. So we will have to keep adding new axioms, but there is no territory to guide us in their choice, so different people will prefer different choices, mutually contradictory.
At the end the set theory research will be fractured into hundreds of competing definitions, all underspecified. The statements will have to be prefaced by ever longer "according to this collection of axioms" which will make things difficult and error-prone. (Things you will learn under one collection of axioms may be false or even nonsensical under other collections. So you will have to re-learn everything over and over again if you switch to a different system.) And the preferred collection of axioms, which will most likely provide you opportunity in the peer-reviewed journals and research grants, will be decided "politically" (as the most popular among the currently established researchers); which according to some people has already happened with ZF(C).