cloud – How do you call a place where one can safely leave a home built computer and connect it to a high-speed Internet?

I found out today about AWS’s and other cloud providers’ outbound data throttling with huge fees for anything above 10GB and I wondered if I could use my own home built computer as a private cloud. It has everything except a high-speed connection, so if there was a data-centerish place, where it could be safely left connected to the web that would solve so many problems (assuming it wouldn’t cost an arm and a leg as well…).