Is this possible?

Probably not. The old Dell would have a power supply able to handle the original configuration, but little else. By the time you replace everything, you may as well build new. I would suggest filling in the template in the sticky "How to Ask for New Build Advice." Depending on what parts you already have, expect to spend a minimum of $450, and more likely $550.

If you *really* wanted to you can. However, it won't be cost-effective, since you'd likely have to replace pretty much everything (and I mean everything) so it would simply be better to provide yourself a budget and build one from scratch.

depends on your defination of "old dell desktop"

You maybe could... It depends mostly on the power supply and whether it can dish out the power to whatever you want to put in it, and whether the BIOS is locked to the original CPU frequency/multipliers (and/or supports the processor you want to use).

I personally wouldn't do it unless the thing is under 200$ and relatively modern (DD3, socket AM3/2 or the intel SB socket, Sata2 or better, etc), all you are really getting is a proprietary Dell Case/Mobo/PSU (all of these could end up being useless depending on dell's design choices for that PC), some RAM (also useless if it's DDR2), an HDD, and an optical drive (limited value if they are IDE instead of SATA and you end up needing a mobo).

All of that stuff could be bought for ~300-400$ without all the potential gotcha's.