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There is no matter.

It's cold there because there isn't anything there.

So there is nothing to conduct or convect the heat away.

It's like a giant vacuum insulated thermos.

Is putting data centers in thermos' a good idea?

i am not saying its a good idea, just wondering because you say space has no temperature, but that makes no sense for the reason CMB radiation would prevent you from having 0 k right? and in fact how would you even measure it? wouldn't the measuring device its self have way more then 0K?

plus you would have to insulate the servers from the sun...then have radiators like the ISS... i think its just way easier to run a server on the ground

Why did you say "eventually" in your original post? That answers your question.

It's a rate problem

Yes and no.

If you had a thermometer that had no heat generation then yes.

If you have a resistor or other heat generating circuit then you need to have the needed surface area to radiate the heat away. If you don't, it will heat up. It's a rate problem.

what thermometer would you use to measure the temperature of space?

Thermocouples