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No, you cannot have a site 24/7 that is undetectable. It would take a immense abount of hacking and log hiding with the ISP to do so, and would probably be a full time job maintaining that cover.
I'm surprised no one thought of this: a reverse proxy server. Traditionally proxy server are used by individuals to hid themselves from servers, as a form of privacy. Well the same can be done in reverse: to hid the true source of the server. If you type "Reverse proxy" into google you'll find tonnes of references. As for tracing a person down to their physical address, this gets into the realm of court warrants and such. An ISP will know which subscriber will have what address via their DHCP, but I doubt they'll give it out without a court issued warrant.