If I'm understanding you correctly, you are saying that you would have the computers connected to the network, using their hd space for backup.
Several reasons come to mind:
Reliability (especially the hard drives--how long will they last?)
Size--some older computers have small hd's which might make managing backups difficult
Management--additional management complexity due to having the backups distributed across different machines.
OS Limitations--if you have a machine with lots of storage and have several computers backing up to it at the same time, you may run into issues with the number of simultaneous TCP connections. XP Home is limited to 5, XP pro is limited to 10, 2000 (I think) is limited to 10 also.
Sorry, sir, but I don't think this is a great idea. If you wanted to use just the hard drives themselves by putting them all in one computer or into your server or by putting them in external hard drive enclosures, that would seem to be a bit easier to manage and use, but you still have the fact that hard drives are likely to fail after several years.
First and foremost, you need a reliable backup system. Don't jeapordize it simply to make use of old computers. You cannot afford to have your backups fail when you need them.
Also, you might want to think about using multiple external hard drives for your backups and rotate them, keeping one of them offsite in case there is a fire or something like that. (It's not nice to think about, but if that happened, you'd not want to lose all your company data in addition to the physical damage.)