[AstralStorm]

How do you cool them in space?

[Gravityloss]

Radiators. Need pretty big ones if the computers want to run cool.

Probably would be better to manufacture most of the stuff on some asteroid or on Mercury and have the computation radiation shielded by a significant mass.

[fatjokes]

Do you need to? Without enough gas in space, how likely is it for something even as powerful as a data center to overheat?

[thinkcontext]

You have gotten it exactly backwards. Vacuum is a great insulator, getting rid of waste heat is a huge problem in space.

[YUMad]

What do you mean? The lack of 'gas' aka material to dump heat into is exactly the issue.