I do stuff like this once in a while. I just set up Audacity to record from the Stereo Mix, make it record, then play the video or song I want an MP3 of. When it's done, you stop it, trim it, and save as an MP3. There may be qualities issues with this method, but I've never been too hindered by it.

Of course, you can't have anything else on your computer make noise while what you want is playing, lest it get recorded as well.