Running Civilization 5 Pitboss in the cloud

I use a dedicated (desktop) machine for running a pitboss server, and that would seem to be the intended use case.

In theory you could instead pay for some type of dedicated server in the cloud, although I am not aware of any servers that specifically support Civilization V. It may be difficult as it is a Steam game, which means that the server would likely need its own Steam account with a copy of the game. I am unaware of why you couldn't run it on a cloud instance, although I would confirm you can meet the RAM and DirectX requirements, and if the game is in fact compatible with the Operating System.

Depending on your goals, a final alternative is to do a play-by-email game. This would mean not playing with simultaneous turns. Although Civilization V does not directly support play by email, some solutions are posted in response to this question. As discussed in the linked question, the first solution is to do hotseat game and email the save file manually. The other option is to use an external service to help, with Giant Multiplayer Robot being relatively popular.

You can use a cloud compute provider and remote desktop if you want. For example, I'm running a game on a GCP instance (2 CPUS and 7.5GB RAM) after following these instructions, and just have to take my turn by logging into the remote desktop. That setup is a little slow (so I do everything in strategic graphics mode) but it works and is projected to be $55/mo, which gives you almost 6 months with their free $300 for signing up.