There's a lot of ways to do this, you can run your own box (I do this with a Beaglebone sitting on my desk), or pay any number of internet people to run it for you (I do this too). Big names in Virtual Private Servers (VPS)
- Amazon EC2 (Yeah I have one of these.)
- Linode (Yeah I have one of these too.)
- Digital Ocean (Uh, yep this one too.)
We're using Digital Ocean, the steps to spin up a server will differ with approach but interacting with it once its up is pretty similar.