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> Cooling is the biggest reason for space datacenters, heat is movement of particles and vacuum being an absence of particles, so it's by definition cold.

That absence of particles also means it is a nearly perfect insulator, with almost no heat transferred from the station to space by contact with space. No convection either.

That leaves cooling by thermal radiation, which is not a very good method.

Space is indeed very cold, but it does not cool you off very quickly. The lack of particles means it's harder to get heat away from yourself. Essentially all of the energy produced via solar panels would be converted to heat by the computers.

It's cold in a sense that is not very relevant. Your tumbler has a vaccuum layer because vaccuum does not transport or absorb any heat. you need those atoms to carry away heat.