How are atoms entangled and can it be done remotely?

[DrChinese]

What are these severe constraints?

You would need to be more specific, as I mention. A hypothetical scenario:

There has been discussion of performing Bell tests on photons at large distance, perhaps even as far as the moon (using a mirror left there). That would show long distance entanglement. It is possible to store the entangled state of a photon for a period of time in a certain kind of lattice structure (IIRC), and then retrieve it. So perhaps that could be placed on the moon. Then you could have 2 entangled things far apart.

But that is probably the easiest of all the scenarios.