Since the 4 hands are all equal multiples, you should only need two shafts and a bunch of 16:1 reductions, like couples of 10 and 160 teeth. Second hand 10-tooth gear drives a stub 160, coupled to a 10, which drives 160 on the 4-minute hand, coupled to a 10... In my head, gear trains get more complicated when they need more shafts, because making sure all those different shafts are aligned and non-interfering is hard. Just stacking more 10/160 gears on longer shafts is pretty straightforward. Your clock would be deep, but not necessarily large.

If you have a 3D printer, you can download models of gears from places like https://mcmaster.com and print them out. Or modify them and print them. A 3D printed clock isn't likely to last until 2038, but it's great for prototyping.

I'd go with a flip clock so when we get to that day in 2038, the numbers just all flip back to 0

This clock? https://retr0.id/stuff/2038/

It's something I'd love to build. Displaying the time in ff chunks of 255 seconds is genius. It also makes a very readable clock!

Check out Clickspring on YouTube. He's literally rebuilding the Antikythera Machine. Hell, reach out to him, he may be interested in helping you out. If nothing else check out the many videos he's done detailing the clockwork of the thing. All his videos are pretty good. It's interesting as fuck, anyways. Good luck, I dig the idea.