Map on cloud server?

Can a game be ran on a cloud server, furthermore if I need to get log functionality is this possible?

It depends on the situation.
There’s not many platforms that offer it, so if you wanted to release a game that way it’s going to struggle to profit.

I mean, purely from a functional perspective, can it be done easily?

Though it’d be the same as having an online game, by perhaps putting the software and map on a server?

You can use something like LiquidSky which is basically a low-latency cloud computer, you can install your game on there but you have to pay for the time you use it.