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.