The basic components you need are a CMS, File storage, and a player
For a CMS you can use any established technology that you may know, such as Rails, Django, etc.
File Storage: Many options available, again go with what you know, Amazon S3 or similar services
Player: Modern browsers support the <audio> tag which can allow you to build the whole experience inside the browser. If you wish to support older browsers (mostly IE) you'll also need a Flash based player.
The devil is of course in the details, but once you have these basic blocks in place you can get a prototype working that will allow you to upload, list, and listen to music files.