if only we all had a little device that was always on and and connected….

If I'm reading the implication right, you're having a pretty terrible idea. Glossing over what running a server would do to your battery, it would never work because of the routing issues you'll run into.

With IPv6 it would theoretically be possible, but currently with ipv4 and NATs everywhere, your website would almost never be reachable, even with fancy workarounds like dynDNS