Helping people with computers... one answer at a time.
Old machines and old hard disks have a way of accumulating over time. We'll look at a few ways to keep the contents without keeping the entire machine.
I have a similar graveyard, of sorts, and I know of many organizations facing similar issues. Consider yourself lucky that it's just under your desk, these organizations are littering the hallways of their buildings with old and outdated equipment.
There are actually several options, depending on how much effort and perhaps money you want to put into this project.
By far that the simplest solution for what you describe is simply to remove and keep only the hard disk drives. Label them all appropriately so you know what machine they came from, but the drives themselves are significantly smaller than the machine they came in. This also solves a data security issue: if you keep the drive you don't need to wipe it.
I might suggest keeping or having available a machine that would act as a place to install a drive if you decide you need some data off of it. And that, ultimately, is the inconvenience of this approach: if you decide you need something off of that drive, you'll need to install it into a machine in order to be read.
If these are truly older drives, say of the under 4 GB variety, then perhaps the next most simple solution might be to write the entire contents of the drive to a DVD ROM. The DVDs can then be filed for easy access and read on almost any modern computer. (Depending on the potential sensitivity of the data, I'd write two copies of each in case one is damaged or lost.)
If the drives are larger than 4 GB but still a reasonable size say under 20 GB, then another option is to purchase a new external hard drive of 500 GB or larger and simply copy the entire contents of all the original drives to the new larger one. I know it sounds kind of silly but with today's larger disks you can collect together a tremendous amount of information and have it in one central repository. One caveat: I strongly suggest that if you do this you actually get two larger drives and back one up to the other on a regular basis. If a larger drive fails a larger amount of data can quickly be lost. Maintaining a second copy is relatively inexpensive protection.
For completeness, I have to mention backing up the contents of your hard drive to tape. It's considered an industrial-strength backup solution. Tapes can contain a tremendous amount of information in a reasonable amount of space. The downside is that you then need special equipment, namely the tape drives, to read or write tapes. Since it's a common backup solution for larger installations, you might check with your IT department to see if perhaps you already have this equipment.
For what it's worth my solution to a similar problem is simply a pair of large external hard drives to which I'm slowly copying my collection of old data CDs.