My question is always how the hell are you going to cool them. Do you know hard it is to move heat in a vacuum?

With radiators just like with every existing satellite system.

youtu.be/DCto6UkBJoI&t=12m57s

Very large scale datacenters would likely have some nasty fluid handling problems to solve.

Have you seen the size of the radiators on the ISS ? And that’s just what needed for cooling of body heat for 9 people and basic computer and support equipment.

A data center that is actively pumping out massive amounts of heat would need humongous radiator panels.