Short answer: yes can do this, you do need Apple's server software, but not necessarily the Xserve hardware.
Longer answer: Mac OS X server makes this type of configuration fairly painless to set up and run. Probably the biggest thing you have to worry about is your network -- if that's unstable or not configured properly (especially with regard to DNS), you may run into trouble "finding" the server. I can't point you to a simple how-to guide (though I'm sure they exist), but if you just install OS X Server, it pretty much leads you through the set-up process.
Network accounts get you most of what you're looking for, but in order to allow users to move between machines and get at all of their files, you need to keep all their files (including their home folders) on the server. This means the server has to have enough HD capacity to hold everything, and also fast enough to keep up with everyone. With only 10 or so users, a Mac Mini might be able to keep up, but it'd really depend on what the students are doing. For heavy use, an Xserve or Mac Pro makes a much better server.
There are also a couple of options for how to handle users files: with regular network accounts, everyone's files live on the server, and everyone accesses them over the network. With mobile accounts, users' files get copied to the client computer, and then synchronized with the copies on the server. If people use the same computer over and over again, mobile accounts can make sense; if people log into different computers every time, regular network accounts are generally better.
The main thing this doesn't get you is offline backup. You can use Time Machine on a server, but it's really more end-user oriented, and I'm not even sure if it's officially supported for backing up servers. It's also designed to back up to a single continuously-connected backup drive, which isn't really optimal for server use. You can improve things quite a bit with a little extra work, by doing something like buying 3 backup drives, rotating them offsite, and using the Time Machine preferences to switch which one's the current backup target. Beyond that, you're looking at third-party backup software; lots and lots of choices here, but I don't have enough current experience to make meaningful recommendations.
Edit: Apple just released a Mac Mini Server bundle with 4GB of RAM, 2 500GB HDs, and the server software for $999 (retail, I don't know the education pricing). I have some general reservations about running servers on laptop HDs (the only kind that fit in a mini), but if you're not going to load it too heavily it looks like just what you need.