The theory behind a space elevator is: for every altitude above the earth's surface, there is a corresponding angular velocity you have to be moving at to remain in a stable orbit: the nearer you are to the earth, the faster you must go to avoid spiraling downwards. At ~36 thousand kilometers, the ang. vel. to remain in orbit is the same as the ang. vel. of the surface at the equator, so you remain above the same point on earth at all times. This is called the geostationary orbit. For a space elevator, the entire structure must at the same ang. vel. The section of cable moving in geostationary orbit is moving at the right speed, but any part of the cable below the geostationary orbit is moving too slow and will tend to sink towards the earth. To balance this, the cable must extend beyond the geostationary orbit as well: that part of cable will be moving too fast and tend to spiral outward, balancing the forces on the cable so it neither falls to the ground nor flies out into space. However, because there are opposing forces on each end of the cable, there will be incredible tension throughout, which would rip any existing substance apart, and the greater the weight, the more powerful the force. Theoretically, carbon nanotubes could pull it off, being extremely strong and light, but we have no way of producing that amount of nanotubes yet.

With a lunar space elevator, there are pros and cons: lunar gravity is weaker, so we have the substances to build it already. The lunar "day" is the same as a one month, so the same side of the moon is always facing the earth. The slow rotation period combined with low gravity means that the "lunar-stationary orbit" is much farther out, and the cable must be much longer. On the other hand, since it always points to the earth it might be easier to balance out. Is there really enough demand to build something like this right now?

WHAT a RESPONSE. Thanks, Morimous! To me, it just seems like a crazy idea all the way around. But I have to admit that the reasons why people think it's a good idea are clearer now.

Re: is there a market? I wonder what the use cases would be. Would this, for instance, be a way to take the carbon that we sweep from the atmosphere and send it to the moon? Is that the application? Just the entire conversation seems so crazy.

I hadn't thought about it in terms of getting rid of carbon. Not sure if you mean getting rid of carbon by trapping carbon in the form of nanotubes, or by using the elevator to ship it to the moon? Either way it's not very efficient. The direct purpose of the space elevator is to reduce cost. Getting onto or off of a planet or the moon is very expensive in terms of fuel - I think it's because you have to carry all of the fuel with you, which means you have to carry more fuel to carry the fuel to carry the... (sup dawg I heard you like fuel) so it's cheaper to have something to climb against, and then the elevators would be powered by solar energy, like satellites, or by laser beams from the earth. Getting stuff up to the international space station is so expensive right now they have to minimize use of toothpaste. So this would be very useful on earth, but I'm wondering what is so important about getting to the moon. Do they want to mine for rare metals there?

There's a google X-Prize Space Elevator Race since some years ago. It also appears in many science fiction stories. I just love the idea.

I agree with mormious answer but i allways thought about the space elevators just to the geostationary orbit, not straight to the moon.