They're certainly able to, and have the keys, but they're not allowed.
Privacy laws should be created (or interpreted) to provide similar protection for the privacy of our data stored by a service.
But the main point is: an apartment is not an information system. Throughout history, governments have spied on information systems and systems providers will too. Hell, spouses spy on each other, given the chance.
It is simply common sense to limit the attack surface you present by hosting your own mail server - or not to care.
The only people who get "free" gmail, are people with an @gmail.com account, which in these days seems to be an awfully bad approach - makes it hard to switch your email to another provider if you run into issues with the old one.
in my case, alternatives and other solutions are generally too high-effort for me to bother with.
I'd bet most people who know but don't care feel similarly.
Privacy and email do not mix, and maybe never will. Attacking the victims of the NSA spying does nothing to stop the NSA from doing the spying.