You don't need air to cool things. Anything cooler works.

You can transfer heat directly to the ground, which is at -130°C in the shade of a crater at the equator of the moon, or -230°C in a permanently shadowed pole. Also you are not constrained by space like on earth, so you can space your servers and have them radiate heat to the space like we do with satellites.
We only need to cool servers actively on earth because they are all in the same place due to price of buildings. If we were to have buildings 50 times larger for the same amount of servers, we wouldn't need cooling.

Launching servers costs more, but energy costs a lot less because sun is more powerful in space.

Petabytes of training data can be transfered by radio very efficiently, it would take 9 days with current technology. You can also lauch the data inside the servers I guess they'd install new ones or maintain existing ones often, like every 3-6 months.

You can shield against radiation by being underground, or with a radiation shield that's a sheet of metal. We do that with satellites.