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It's doable (and has been done), but is not entirely easy or cheap. Without getting into the orbital mechanics/whys, a "geostationary" orbit around the moon is not available (it exists but is further out than the Hill sphere and not stable). You can park a relay semi-stably at Earth-Moon L2, but still need station-keeping burns. The moon has has a very lumpy gravity field, so any kind of orbit needs station-keeping eventually.

It's just not super worth it.

If you want to look at a mission that did this, see China's Queqiao.

Orbital mechanics and "next to" don't go together particularly well, so it's not quite as easy as popping something up there.

The Chinese have put Queqiao-1 in the earth-moon L2 point which seems to be working out for them, but I guess the Americans aren't likely to be asking permission to use it.

We have DSN already. As for the moon, it is a nightmare to orbit. Its density is very lumpy, which means orbits around it are constantly being perturbed, and that means you need to bring an annoying amount of propellant if you want to remain stable.

Well technical difficulty is one piece. Cost and ROI are a different one.