Given how much engineering has advanced since the days of the Mercury program, could a similarly scaled single-person orbital vehicle capable of repeatedly sustaining 48 hours in orbit realistically be constructed as a backyard/garage project by a dedicated and knowledgeable DIY team?
(Too much for a comment)
You have excluded the launcher but there are still some very important unknowns in your problem:
At the most basic situation the answer is yes. What's needed to keep someone alive in space for 48 hours is well within the range of DIY work by someone sufficiently qualified. You need a pressure hull, 48 hours of air, temperature control and a toilet of some sort.
Your air system can be pretty much off the shelf--a rebreather for scuba diving provides everything you need, just adapt it to blow into the capsule rather than a mouthpiece.
Temperature control can be some white shutters over black panels, combined with a slow roll like Apollo did.
A toilet can simply vent urine and use the yucky Apollo fecal collection bags.
There are three additional aspects to a space capsule, though, and you haven't defined if they're needed.
- Attitude control. Two options here:
1a) Reaction control systems. Now we are getting into some nasty territory. Hypergolic propellants are nasty, nasty things and I would be surprised if your backyard engineers can even get their hands on them, nor are the likely to have the equipment to safely handle them.
1b) Wheels. Slow, but so long as that is acceptable they'll work. Three electric motors simply turning weights will do it.
Deorbit burn. Wheels won't help you here, you have to go with rockets--this could be the same ones used for reaction control, although the shuttle used bigger versions of the same idea.
Reentry heating. Highly specialized materials and I would expect some three-letter guys (look at the various government security agencies--most of them have three word names, normally abbreviated to three letters) to come knocking on your door if you even try to get them. You also have no feasible means to test your design even if you get the material.