Well, you don't specify an OS, so I'm going to assume you'll be running a server-oriented OS, e.g. Linux, UNIX, Mac OS X, NetBSD/FreeBSD/OpenBSD, or similar.
In that case, I'd suggest a minimum of 512M of RAM. The more RAM, the more disk caching the OS can do behind the back of the web server software. If you're planning to do a lot of dynamic content with PHP or something, make it a gig, but otherwise, 512M is plenty.
Of course, to some extent, I'm assuming that this server will eventually serve web pages as well as music, and that most of that won't involve constantly shoving large files. If all you are planning on ever doing is shoving a handful of large files, I used to easily push 2-3Mbps serving raw MkLinux CD images with a file server running on a Quadra 650 under NetBSD with something like 64M of RAM. (For comparison purposes, that's roughly equivalent to a 486 DX/2 running at 66 MHz, IIRC.)
In other words, it depends.