Can Macs automatically log out after inactivity?

A classic problem for a school or other high-use network, actually, and it turns out that there are a couple of ways you can try to address the situation. Probably the easiest, though, is to enable exactly what you're talking about, auto-logout on idle.

To get to that setting - and other interesting security settings to explore - you want to go to Apple --> System Preferences and click on "Security". You'll see this rather big pane:

One thing you could use here is "Require password to wake this computer from sleep or screen saver", since one of the options at that point is to log in as another user (which is nice: the original student still has their apps running. Or, not nice, since if enough students use a system it could end up clogged and bogged down).

A better solution is to choose "Log out after X minutes of inactivity". The default is sixty minutes, but I would suggest that you drop that down to 30 minutes, or even less: if you make it too short, don't worry, you'll hear from the students about the problem! :-)

While you're at it, I'd "Disable automatic login" too, if you haven't already done so: that way instead of the kids being logged into a guest account or similar, each will need their own personal account (which you can manage from the "Accounts" system preference, of course).

Good luck. There are some interesting applications for managing a network of student computers available too. Go to VersionTracker.com and poke around to see what learn about some applications that can make your management task easier.