[ginnungagap]

Another famous example of that kind is the existence of outer automorphisms of the Calkin algebra, which is a simple question about a naturally occuring object which turned out to be independent of ZFC by work of Farah and Weaver.

Topology and set theoretic topology is full of those statements, are there Suslin lines? Are there S-spaces? Is the product of two ccc spaces also ccc? The list goes on

Another one from analysis is a very strong form of Fubini's theorem that was shown independent by Friedman.

There's surely more but those are the ones I could remember right now!