Do you want to deliver live content (IE, an online version of a "normal" radio station), or are you looking to deliver recorded content "on demand"?
Each will have slightly different considerations, but, in the end, your solution will be broken into a few different parts:
1. The media server - this could be your site, or a server you build, or even a service, but (conceptually at least) it is very separate from your Web site. The playback, on the user's end, will be in a media player (which can be embedded in a Web page) such as Real, Quicktime or Windows Media.
This seems to be the big question and is where I'll start to research my options.
2. Your site can be just about anything (so Joomla! is still a good way to go in terms of getting your layout, your organization, etc.), but you will need to figure out how to embed media player objects (there are some that already exist for Joomla!) into modules or content items so suit your needs.
3. If you are going to require registration, or something like that, you are going to want to think about how you prevent your streams from being accessed by unauthorized users.
It'lll be open to all.Figure out your media streaming first, because the rest of your decisions will really flow from that. Are you streaming speech or music? Music will require a somewhat beefier solution. Also, the number of simultaneous listeners that you anticipate will be a big factor in determining the best way to go.