^If you already have webhosting, and a domain name set up, then yes- you can just pop the mp3 file into a directory. Quicktime plugin will automatically play it in the browser. Though this is assuming the person has quicktime installed, and also that they prefer it taking over mp3s in their browser!
But yes if you just toss the mp3 file in a directory like
http://www.myfakewebsiteasdf1234.com/mymusic/mytune.mp3
it will be all set.
There are free hosting services like geocities, and such. But they have a cap on the size of files, and the total amount of webspace. One mp3 file would max out your free webspace, in other words. There are lots of good hosting companies out there.