Thorormr wrote: Relace both wheels so the hubs are off center. Might need to drop a rim size, 26" down to 24" so they'll clear at maximum eccentric. If you can get 3" of travel it ought to hobby horse rather well. Crank might smack the ground and rim brakes certainly won't work. It's an interesting thought but making custom spokes and relacing wheels is a pain.
My camp-mate Peter has been building wheels like that for 20 years. He can probably do it blindfolded. And he does have a bike just as you describe.