If you want to have services on the internet just rent a VPS. Keep in mind though plan to keep anything on the internet very updated, secure, and maintained. Linode is one such provider. There are many others.
If on your LAN then frankly the lowest power thing you can live with as electric cost will be a major part of the cost. I actually have a fairly substantial Media Center computer but also host my services that in my LAN since it needs to be always on anyway. I would not call my my solution low power but it is on already any way. I just made the system capable of both uses.