Does the computer have to currently be in use? If so I may have the oldest computer that still sees some use...
circa 1994
Packard Bell Legend 104CD Supreme
originally a Pentium 60 currently running a 233MMX with a powerleap module
USR Sportster Vi 14.4 modem (so old it was made in the USA)
96MB EDO RAM (shipped with 8MB)
1.7GB Seagate hard drive (boots windows ME), and a 9.1GB Seagate Elite 9 SCSI drive
Voodoo 3 2000 16MB PCI card (primary) ATI Mach32 onboard for a secondary display
Soundblaster AWE32 w 4MB RAM
external SCSI 4x4x16 CD burner
the SCSI adapter is an ISA one from an Iomega Jaz drive
also has a Syquest EZ Flyer drive hooked up to it
When purchased it had 810MB hard drive, 2X Matsushita CDROM (controlled via an Aztech sound card), this was back before standard ATAPI drives were seen). Forget about pipeline burst cache, this thing has asynchronous write through.
So this thing still works. I actually played Quake 2 on it on Thanksgiving (and it runs surprisingly well with the Voodoo 3). The cards take up all but one PCI slot so further upgrading is technically still possible. Its not terribly slow either, I elmininated a lot of extra stuff that didn't need to be on there and it works. I did some workarounds to make the original Packard Bell Navigator and its flyscreen work in windows ME, as well as the original screensavers.
It was upgraded from Windows 3.11 to Win95, to Win98, to WinME, and yes I'm going to attempt to make it run XP, but that will likely have to be a full clean install.
Its more about the novelty of it working than actually seriously using it, though if you had to go online with it, or type up something its not slow to the point where you'd be hitting it because it is so slow.