How to recover an iPod Classic that won't turn on and doesn't appear as a drive?

Assuming that you've tried resetting while the iPod is connected to a power source, there's really not much else that can be done. The Apple Store, or an Authorized Apple Service Provider, is pretty much your only option at this point.

Also, how long has the iPod been doing this? If it was subject to extreme conditions for an extensive amount of time, or if it suffered liquid damage, it may be broken beyond repair (or repair may be cost-prohibitive, assuming it's no longer under warranty).

Here's something that worked for me:

1. Put the iPod into a ziploc bag and seal it tightly.
2. Put it in the freezer overnight.
3. Take it out in the morning.

I thought that it was just an urban myth, but as the iPod was apparently dead, why not try it? I was (pleasantly!) shocked to see it start up again the next morning.