This, I now sheepishly realize, makes a lot more sense than what I'd been picturing. So the server... Is not just sitting in Taluntain's house somewhere?
Indeed not. Our new server is located in a big private datacenter in Chicago. To get the kind of connectivity that we need for fast world-wide access, hosting the server locally is not really an option.
As for what that looks like, I'm posting this one because it's rather Christmassy:
In reality, it's usually more like this: