Now, I have no idea how practical it is to build one (Angela Collier has a video saying it's kinda ridiculous), but it's a cool idea.
https://www.youtube.com/watch?v=Z5aHMB4Tje4
Also since rockets have moved away from hydrolox, it would be nice to have a greener launching system.
Build towers around the equator. Build a ring around the equator on those towers. Build a ring inside the ring, maglev supports. Evacuate the ring, spin the inner ring above orbital velocity. The objective is to generate as much outward force as the weight of the entire assembly including the towers.
Do it again, on top of the first one. Keep doing it until you reach synchronous orbit. If you want to go higher the inner ring is not moving, exerting downward force countering the outward force of the rest of it being above orbital velocity.
Forces:
1) Compression on the towers. Note that this goes to zero as tower height goes to zero.
2) The outward force on the ring exists across the whole ring, but the downward force of supporting the towers only exists where there is a tower. Your ring needs to be stiff enough to counter this. But, again, note that this force goes to zero as the space between towers goes to zero.
3) Maintaining a very hard vacuum in the structure.
We know how to do #3, the others must have answers. Thus it can be built.
Nothing else is feasible with current technology, the taper of the cables is highly dependent on the strength of the material (in tension, an elevator is based on always pulling out as the upper end isn't hooked to anything.) You need much better than anything we can currently do before the taper blows up so badly you can't build it.
There are other concepts like space fountains, orbital rings and sky hooks that seem more doable -- especially the sky hook seems close to do-able, especially on the Moon.
Orbital rings--only if you have elevators. Remember, the Ringworld is unstable. So is every other planetary ring.
I do not recall numbers on hooks so I will not address them.
The Moon has a whole different set of problems. There is no synchronous orbit, elevators must go above synchronous orbit, so the normal version can't exist. Nor can anything stand up to be yanked around by the Earth.
But there are two cases that avoid the yanking problem: pointed towards and pointed away from Earth. Current cables are good enough for a useful Earth-pointing cable. The free end dips below synchronous orbit, but it's moving very slowly. You do what people think rockets do--go up. It takes a lot less energy to catch the cable than it does to even reach orbit.
How to have such a cable in an environment with geosynchronous satellites is another matter...
There's also another interesting cable situation. Cable on Mars? Iffy--and those two moons would be a major problem. But flip the problem over--put the cables on the moons. The low end dips into the atmosphere at aircraft-type speeds. The cables can toss to each other. The high end can capture/eject to Earth or the asteroid belt.
Space elevators only (theoretically) work because the entire structure is in tension. And the only material we currently know of that can handle the tensile forces is carbon fiber.
