Spotify?

[Moonliner]

To start with Spotify has the most complete music library of any online service since Napster. Not everything to be sure, Disney soundtracks are conspicuous in their absence for example.

With a Premium (paid) membership you can plug in all the songs/albums/artists you like and listen to them all day/night long on your PC, Smartphone, and some other devices (like Onkyo receivers).

Past that you can share the lists you have created with others you know.

As GD says, Spotify does not seem to have a "Radio" function that lets you listen to random songs, however there are third party sites like topsify.com that it seems are trying to fill that gap.