The ground would probably work fine as a heat sink.

It's already pretty cold

Go Thorium MSR and bury it underground and you don't really have to worry about it. Might need some modification for moon gravity but otherwise seems like the best bet.

https://www.thmsr.com/overview/

That was my first thought, but then my second thought was even more terrifying - how do you protect your nuclear power facility from celestial impacts? The moon must get pelted with thousands of little bits of space debris every day considering it has no atmosphere. All it would take is a basketball-sized meteorite to slam into the reactor chamber and possibly cause a meltdown.

That's a challenge that people are working on for sure. Likely some kind of radiant cooling, but it's a lot of heat.

Heat also dissipates via radiation, not just conduction. I would imagine that nuclear power on the moon won’t involve hauling a lot of liquid coolant/heat exchanger/energy transfer because liquids are wicked heavy, hauling that up to orbit and then landing it is gonna take a lot of energy. They do acknowledge that cooling is an issue they’re working on.

Maybe some kind of RTG? I couldn’t find an article that said what the NASA contractors chose to build.

Not really. Current battery technology is to put it lightly not the type of thing you want to rely on for long term life support. Lithium ion the current go to for rechargeable batteries physically degrades as you charge it. One of the main things you can do to reduce this is don't fully charge the battery. For example if the battery degradation from 0%* to 100%** is a cycle then 50% to 80% is only 21% of a cycle. That'll extend the lifetime of the battery (not the capacity) by about 5 times! That's pretty significant but you lose out on 20% of the batteries capacity permanently, even as the capacity decreases from degradation.

You've probably seen the hype about Sodium batteries which are currently 50% less energy dense which just immediately means NOPE for use in space.

* Lithium ion batteries are extremely difficult to actually fully discharge (controller won't let you)

**Lithium ion batteries should never be fully charged it causes them excessive damage so the controller prevents this from happening

Much less power for the weight, and weight is the big deal when you're sending things from earth.

Just for fun, go check out xkcd's new What If video. They go into heat dissipation of a nuclear reactor in space (not the moon, but still incredibly interesting, informative, and entertaining).

You don't just turn a nuclear power plant off.

If sun is an issue, it might be simpler to set up a “beach umbrella”

Ehhh, screw the whole thing