I had to laugh at the TRS-80 or Compaq or whatever antique computer he was using
There's actually several ways you can layer multiple synthesizers and most of them are really quite simple.
First of all, if all of your gear has standard 5 Pin MIDI ports, you can daisy chain them from OUT to IN to THRU to IN and send the same control signals on the same channels to all your modules or synths in the loop.
Likewise, you could spend a few hundred bucks on an old fashioned MIDI Router and have a little more control over where your output is going.
Secondly, if all your gear has only USB MIDI ports, then you can use a computer with Freeware MIDI Routing Software to control where your output is mirrored to and everytime you hit a key it will play simultaneously on whatever patches you have set on your other keyboards/modules/soft synths etc.
If you have a mix of USB only and MIDI only gear you can still use the USB method using a good multi port USB MIDI interface.
The only other really required piece of gear is a decent mixer to mix down all the audio from all of your sources into a single stereo output.
For starters, you could use a single hardware synth and a couple of soft synths to create some pretty complex layers without spending too much cash.
Other than that, the sky's the limit, or at least your budget for rack modules, synths, effects pedals and mixers.
Actually I'm sitting at my studio right now and everything's powered up so here's what 5 hardware synths and 2 software synths layered up sound and look like on my computer.
It's a little freaky, but actually sounds pretty decent.
This is total overkill though
Keyboardforums won't allow me to upload an MP3 so I put it up on my Soundcloud account.
https://soundcloud.com/happyrat1/mad-layers
Enjoy
Gary