Music that sounds like someone else’s could be construed as plagiarism and could land you in court, likewise using music without permission is really a no-no, even if it isn’t for something commercial. Although you won’t be making money directly from a demo reel your prospective employers won’t be impressed if you use stuff without permission.
One thing you might try is asking an up-and-coming composer if you can use something. You may find that they’d value the exposure if properly credited. After all, your reel will be considered by just the studios that may be looking for music too. I don’t know, but perhaps that is how Steven Chen got the excellent music for his reel. It’s by Justin Durban .
Another alternative that you might consider is to get someone who has an orchestral sample player to use a royalty-free midi file to output some music. The results can be pretty much as good as hiring your own orchestra.