Pluto and its moon Charon are tidally locked onto each other, like Luna is to Earth or the Galilean moons are to Jupiter, only it goes both ways. The permanent sub-Pluto point on Charon is opposite the permanent sub-Charon point on Pluto. So, if humanity ever colonizes that system, we could build a chain or cable between them if we chose to. And if we had a reason.
Regarding space elevators, I’ve heard it said by people in a position to know such things that a cable made of buckminsterfullerenes would be strong enough. In Kim Stanley Robinson’s Mars trilogy, there’s an elevator/cable at Mars, and several around the Equator at Earth. In fact, space travel is so common by the 2100’s that Earth’s cables are Y-shaped; they split near the bottom and end at stations straddling the Equator.
The climax to Red Mars, the first book in the trilogy, is a spectacular event involving Mars’s space elevator, but I won’t spoil it here.
There is nothing that you could anchor the chain to on Earth. That part of the ground would just rip out as the Earth rotates. (remember we’re talking about the angular momentum of a 6x10^24 kg mass rotating at almost 1,000 mph at the equator…ain’t no bedrock anchor gunna hold that)
ditto for the anchor on the moon upon which you would hypothetically pull to change its orbit.
Connecting the chain would not be a problem. Simply cover the Earth and the Moon with a mesh made out the same stuff as the chain. The chain would then just connect the two meshes together. Piece of cake.
We had a colloquium last year by one of the guys who’s talking about a fullerene space elevator. Strictly speaking, it’s not buckyballs, it’s nanotubes: Same idea, but long instead of round. We currently have the technology to make them in lengths of a few centimeters, but it’s very slow. If I recall correctly, the tether would need a taper ratio or 5:1, meaning that it’d have five times the cross-sectional area at the maximum as at the minimum, and steel, by contrast, would require a taper ratio of 300000:1 .
As for the chain to the Moon, why not just anchor it at one of the Poles, and put a fishing-lure swivel on one end? That way, it wouldn’t wrap around the Earth, or rip off chunks of rock, or anything unpleasent like that;)