Yes, it is possible. I know people at Michigan Technological University that have made computers out of smaller things, however, you have to order specialized items. When you think about it, laptops are pretty small, and you could fit laptop components into a PS2 case.
First off, if you wanted to do this, you might have to find a different way to store memory, rather than a bulky hard drive. There are memory storage options such as memory that is like RAM, that uses the same type of chips, but they function like a hard drive... but they're much more expensive of course. It just wouldn't be worth it. However, if you had money to burn, you could make a computer work in a PS2 case... it'd just take time, money, and patience to find the right things.
Intel (I think it is Intel) makes a really small mobo, called BTX (I think it is anyway, it may be ITX or something completely different). But this board is super small, and could fit in a ps2. You can do it man, just gonna take some time and patience. And lots of money.
I was thinking of puting a computer into a briefcase But a vcr might be pretty crazy as well. Personally, if I want to build a mini PC, I would like to put a screen and keyboard and every thing
but anyway, about the PS2. I can fit, but I think the PSU hangs out a bit. Also, you would really need to find a really really small motherboard