It is surprising, but physicists don't care much for set theory. Heck in their daily work they still use infinitesimals!

So yeah, among the few physicists who understand enough set theory to comment on this, most of them are totally fine with admitting that our reality is not based on set-theoretic real numbers.

For mathematicians, sets are the base reality and everything else is constructed in terms of those. For physicists experimental validation is the base reality, and if it disagrees with set theory that's okay with them; it just means that the set-theoretic real numbers aren't the correct real numbers for doing physics. They're very pragmatic when it comes to mathematical foundations, and that's probably the right approach to take for doing physics.

There have been a few attempts to formalize real numbers that more closely match our observed reality; look into measurable sets. Most of these require a large cardinal, so they go beyond ZFC.