Legally the patent has to disclose to someone skilled in the relevant art, how to enable the invention. So basically, if you handed this patent to engineers at Boeing, NASA, SpaceX, MIT, or UIUC or whomever, could they actually carry out the invention.

The written description also has to be sufficient enough to show that you actually possess the invention when the patent is filed.

So technically, if this can't be built today because we don't know of any materials that can actually behave this way, it's not a valid patent.

I'm not totally sure what happens in cases where the invention is possible but so prohibitively expensive that it's not really possible. I'd imagine that is valid.

Traditionally, invalidating a patent based on written description or enablement issues was hard. But the Fed Circuit has made it easier in recent decades.

I'm pretty sure it wouldn't be valid, but I'm not familiar with the sort of building materials necessary to build a space elevator. My understanding is that nothing we have today could do it, but that's based on light reading.

EDIT: I'd also be willing to bet that some popscience mag or text book discloses this general idea anyway, so it'd be invalid based on prior art too.