The few times I have seen ancient hardware still in actual production use, it's usually been virtualized to run on more modern hardware, because if it's mission critical, it's hard to have any faith in an ancient piece of hardware working reliably. These places aren't the sort to have staff that can replace a blown capacitor on the motherboard, or even splice a modern power supply into the old style connectors when the power supply fails. Or where do you get a known good 120MB (that's MB, not GB or TB) hard drive because the hardware can;t address a larger one, even if the interface is compatible?
I have, within the last 10 years, come across an environment where Windows NT 4.0 was the operating system - driving a critical control software with a significant life safety factor (it controlled the electric arc furnaces in a steel mill - tripping the gate open at the wrong time could flood a working area with molten steel, or failing to stop the crucible dump at the right time, etc.). Why was this not upgraded? The new software, which would run on a modern system with a modern OS, also required all new control hardware, an investment of well over 1 million USD to replace. Combined with whatever losses would be incurred during the complete production shutdown to replace all the control hardware. It worked, so it was left alone. The only concession was that it was connected to an air gapped network so there was absolutely no way to externally hack into this system (which with the ancient OS would be otherwise fairly easy to do). We also have a pipeline company as a client who also takes great pains to control what connects to their publicly connected network, let alone the air gapped network for the control system. Even the monitoring system is completely independent of the control system. There may be some old out of date systems running on that control network there as well.
But those are exceptions. I'm not real interested in "near-vintage" computing - 386, 486, early Pentium machines. I do have some older 8 bit and early 1 bit machines, those run, like James said, period correct operating systems, but they aren;t production machines. If I did have a 486 or Pentium machine - those too would run period correct stuff, not some cobbled together attempt to run something more modern. The oldest machine I have, though just shelved, is a second generation Core (the one after Core2 Duo) which was happily running Windows 7 when I replaced it. ANd I forgot my model railroad control machine, which is a second gen Atom dual core running Ubuntu - something like 10.04 I think, whatever was current when I built it, which was 8 or 9 years ago. It's been stored for the past 5+, since I moved. I know a new version came out shortly after I built it, but since I had to custom compile my own driver for the wireless card I didn't feel like upgrading and doing it all over again. It does what it needs to do with the old version it has, though I'm tempted to see how well it cna run Windows XP since my new control system will run a VB app.