Chaining bodies

[Deshara]

I was writing out a setting overview for a story I'm writing, and though all of the technology in it is fairly beleiveable (No warp drive, lasers, teleporters or anything) for what is essentially a space opera that I've been told is like an end of the world-ish version of FireFly (Never seen it, so idk), there is one thing I'd like to check: What would happen if a chain or something were attached between two astrological bodies, like the earth and moon, or two different planets in one solar system..?

[Cap'n Refsmmat]

The biggest problem I can see with tying a chain between the Earth and the Moon is that the Moon revolves around the Earth at a different rate than the Earth rotates, so the chain would end up being wound around Earth and dragging the moon closer.

 

That is, if you could build a fantastically strong chain big enough.