The Stefan-Boltzmann Law tells us that radiative power scales to the fourth power of temperature (T^4). While terrestrial cooling is largely linear and dependent on ambient air/water temperature (the "wet-bulb" limit), a radiator in space is dumping heat into a 3-Kelvin sink. That thermal gradient is massive.

This is misleading: - A radiator only “sees” 3 K if it’s perfectly shielded from the Sun, Earth albedo, and Earth IR. In Earth orbit you can easily get hundreds of W/m^2 incident; without sunshields the net rejectable heat is greatly reduced. - You have a "massive" advantage only if the radiator is allowed to run very hot: At 300–310 K with \epsilon \approx 0.9: about 400–500 W/m^2. Effective "radiative heat transfer coefficient" at 300 K: h_rad \approx 4\epsilon\sigma T^3 \approx 5-6 W/m^2K. That's orders of magnitude lower than forced convection in air (\approx 50–500 W/m^2K) or the water side of a heat exchanger (>=1000 W/m^2K).

The thermal gradient in space is meaningless because there is hardly any matter to dump the energy into. This means you are entirely reliant on thermal radiation. If you look at the numbers given by Stefan-Boltzmann law you'd see that means to radiate a significant amount of energy you need a combination of a lot of surface area and high temperatures.

This means you need some sort of heat pump. For a practical example you can look at the ISS, which has what they call the "External Active Thermal Control System" (EATCS), it's a complicated system and it provides 70kW of heat rejection. A datacenter in space would need to massively scale up such a system in order to cool itself.

Stefan-Boltzmann is about absolute, not relative temperature.

When one does the math on the operating temperatures of regular computing equipment that we use on Earth, how much heat it generates per watt, and how fast it would need to sink that heat to allow for continuous operation, one gets surface areas that are not impossible, but are pretty on the high end of anything we've ever built in space.

And then you have to deflect the incoming light from the Sun which will be adding to your temperature (numbers published by private space companies regarding the tolerances of payloads those companies are willing to carry note that those payloads have to be tolerant of temperatures exceeding 100° C, from solar radiation alone). That is doable, you could sunshield the sensitive equipment and possibly decrease some of your thermal input load by putting your craft out near L2 which hangs out in the penumbra of Earth. Still a daunting technical challenge when the alternative is just build it on the planet with the technology and methods we already have.

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