Could we put data centers in space?

What a crazy idea.

Cost to orbit is huge. Power availability will depend on pushing enough solar up there. Data centres use lots of power.

Getting rid of excess heat is a major problem in space and data centres generate lots of it. Terrestrial cooling is trivial in comparison.

Hardware obsolesce will be a major hurdle as will repairing stuff that fails.

Radiation is a significant problem in space. Shielding will be very heavy or the hardware will have to be build tolerant which is much more expensive and results in a much slower solution.

Much easier and cheaper to install lots of solar and batteries in the middle of a sunny desert somewhere really cheap and build your data centre there. Then build fibre connections to wherever.