Space Elevator

Regarding actual space elevators though, while they're not sci-fi to the extent of something like FTL travel - ie. they're technically not physically impossible - they're still pretty firmly in the realm of sci-fi. We don't have anything close to a cable that could sustain its own weight, let alone that of whatever is being elevated. Plus, how do you stabilize the cable and lifter in the atmosphere?

A space elevator on the moon is much more feasible: less gravity, slow rotation, no atmosphere, less dangerous debris. But it's also much less useful.

This is especially true considering that you don't need something that barely holds - you need something that you know will hold up to many times more weight than it needs to, so that it can be safe: the potential energy such a thing would store would be enough to dig into hundreds of meters of rock all around the world, if it ever crashed. So, you have to ensure there is no realistic chance of it ever crashing. It also has to be highly non-fragile in other ways, so that a madman with a bomb or a freak collision with an airplane or a meteor (especially likely in the thin upper layers of the atmosphere) won't bring it all down.

This combination of properties may well be completely impossible to actually achieve in a material. Even if there is no obvious basic law of physics that it would break, that doesn't mean that it wouldn't break other, harder to touch, derived laws.