If the sun can radiate heat into space, why is this a problem for data centers in space?

Is there any way some of the waste energy can be stored in an electrochemical cell / battery or capacitor, and then reused? Sort of like regenerative brakes in an electric car. A thermopile, perhaps?

Mr Carnot tends to say no. At least not in any useful manner.
The actual work that can be performed by the heat is firmly bounded by the thermodynamic efficiency. It is bounded by the two temperatures, the hot side (your heat) and the cold side (your heat rejection system.) Which just gets us to where we started.

\eta_I = 1- {T_{cold} \over T_{hot}}

Whilst the cold of space is about 3 K, and might give one hope, that is not the cold side temperature you are dealing with. You need a physical cold side sink that is colder than your heat source in order to do work. And how cold your cold sink temperature is is limited by its ability to radiate heat away. It will reach a steady state temperature when it is radiating heat into space at the same rate as you are generating heat. Since that rate is proportional to the temperature to the fourth power there is a brutal reality that limits how cold you can get the radiator. And it isn’t good. (It seems that realities of heat transfer within your system mean the actual transfer rate is closer to temperature to the third power, but this doesn’t actually help.)

The cold sink side is never much colder than your waste heat source. Thus efficiency is pitiful. Tiny fractions of a percent. If you limit the heat going in, the cold side could be colder, then efficiencies rise, but since you had to curb the energy going in, you don’t win.

The obvious real world example is RTGs in spacecraft. They have a radioactive heat source that can reach temperatures near 2000 K. And yet they have efficiencies of around 5%. Trying to get anything useful out of a heat source that is of the order of 350 K is essentially futile. Not actually impossible, but compared to the energy being gathered from your solar panels, totally useless.

This is why it gets called low grade heat. Good for heating things that need to be warm, but not much else.

I had a Stirling piston where you could put a little tea candle underneath it and it would produce 1/4 watt, easy. You could power not just one, but two or more dunking birds. Loads of useful power!

I understand that NASA is pursuing this exact technology, except with more plutonium: Dynamic Thermal Energy Conversion - NASA

This was obviously in jest, but the roundtrip time for a message to Mars is 6 to 44 minutes, depending on orbits (and at some point Mars must be be blocked by the sun). So, not exactly lighting response time to queries.

I don’t really understand this. Explain here, please.

At the basic level, heat is atoms jotling each other with energy. The faster they hit each other, the higher the temperature. (At a certain point, they are vibrating too fast to stay in the lattice of solids, so the substance “melts”. Add more energy, the atoms fly off into space until they hit something, and this is a gas.)

No atoms, no temperature. There may be radiation (electromagnetic waves, i.e. photons; or sometimes other subatomic particles) but this radiation does not produce heat unless it hits an atom and gets it moving faster.

I think it’s worth noting that there is no perfect vacuum in space. A perfect vacuum is considered impossible; quantum physics dictates that there is always something. But outer space gets close enough.

As for how empty space is near us…

In the Solar System, space contains on average five atoms per 1cm3.

Apollo capsules rotated so that the area exposed to the sun would turn away from the sun and lose heat by radiation. Despite the data center in orbit producing its own heat, is that method in any way applicable?

Not a clever strategy. If the satellite is generating internal heat, and that heat is conducted by a solid medium (rahter than pumped by liquid cooling) then the part facing the sun gets heated and the same medium that pulls away heat will now conduct the sun’s heat into the computer onboard. An alternative strategy would be to have two computers, and only run the one (creating heat) whose radiating unit faces away from the sun. Rotate, and switch to the second computer.

Simply shielding the “radiator” fins behind the solar panels seems like a better strategy.

If you had a hot object on the space station, what would be the difference in the time to cool down if the object was in the space station versus outside of it. For instance, an astronaut microwaves a couple of Hot Pockets for too long. They are burning hot. He’s really hungry and wants to eat as soon as possible. One Hot Pocket he leaves in the galley and the other he releases into space. Which one would be the first to cool off enough for him to eat it?

The one inside, in contact with the air, and it wouldn’t even be close.

How much longer do you think it would take for the outside one to cool down compared to the inside one? Like, if the inside one took 5 minutes to cool down enough to eat, how long would it take for the outside one to get to that same temperature?