When it stops, I tried to get it restarted, but it doesn't even act like it wants to start
A bit more detail would help the diagnosis. Does this mean that it doesn't turn over at all, or that it turns over but does not "catch" and start?
If it doesn't turn over (i.e., the starter doesn't do anything when you press the start switch), then it's almost certainly an electrical problem. If I had to guess, I'd say probably a bad ground connection somewhere. The fix is simple but time-consuming since you have to take apart a fair bit of the tupperware in order to get at the connections.
If it turns over but doesn't catch, it could still be electrical, but might also be fuel or compression. When it dies and won't start, try kick-starting it. If you don't feel any resistance when you kick through, that suggests that you don't have compression. Possibly a sticking intake or exhaust valve or just a piece of carbon on a valve seat, but that's a long shot. Since it's an intermittent problem, the first thing I would try is to run a few capfuls of SeaFoam into the carb throat, the put in enough to kill the engine, let it sit for a while, then start it up again.
Since you can start and get a block or so away before it dies, my guess as to the *most* likely problem is a fuel valve (vacuum actuated petcock) that is sticking shut now and then.