You can store files in a BLOB field in a database for example. Another possibility would be to store them directly on the file system, but it has its problems (such as the lack of metadata).
Then you'll need to figure how you want to map each file, for example http://myserver.com/Artist/Album/Songname
or http://myserver.com/listen?uniqueid=743
.
File upload servlets are available all over the web, so that shouldn't be a problem. The rest is up to your programming skills (this might be too big of a project for you).
Then you'll just need to connect those things together, and that's the part where you start to program (or hit the Google hard).