A lunar space elevator will never happen. First of all the gravity is unstable.

And even if you do so, the main advantage of an elevator can also be had with an accelerator, there is no need to build an elevator. Because there is no atmosphere you can simply fling stuff into orbit for pretty much exactly the same price but far, far, far, far less investment.

The one place elevator actually makes sense is Mars potentially. But that is a very long way off.

Even lunar space elevator will cost an insane amount of money (and likely still not possible in this century), and it will be irrational to build it just to get some cheap fuel into orbit.

And even then gas station in Lunar orbit is useless. If there is a need to refuel, you will simply get fuel into orbit directly from whatever container in the elevator to the ship nearby. There is zero point in inserting a proxy station in the middle of this process.

However we juggle the possibilities of space travel, in 99% of all schemes gas station in space is useless.

Doesn't the moon rotate too slowly to make a space elevator work? IIRC you can't take up a ground-stationary orbit around the Moon because the Earth would constantly knock your satellite around with its big gravity well.

You can have a ground stationary satellite if you tether it to the moon. Though obviously the tether has to deal with that extra load.

The best sites for permanent outposts are the poles though, it would be quite a drive from the equator.

I think this is where the Lagrange points L1 and L2 would help.