I ran one off of very little running, so not quite zero training. I signed up for the Long Beach marathon in June of 2010 (the race is in mid October), and broke my heal shortly after this. I didn't bother cross training and had basically written off the marathon as a loss, but after running 2 miles without pain about 4 weeks before the race, I decided I'd still jump in and shoot for a BQ (stupid idea). I ran weeks of 15, 27, and 70 (yes 70), then took off almost a week right before the race because that 70 mile week just about wrecked me.
So I jumped in the race and got out a little too fast. I was on 3:05ish pace through 10 or so and started feeling really good, so I picked it up and was on sub 3:00 pace through 16. During mile 17 I started feeling pretty lousy, and at 18 the cramps began. My calves, hamstrings, quads, hip flexors, arms, shoulders - you name it, it cramped up. I was reduced to a slow jog and stopped numerous times to stretch. I ended up finishing in 3:26:25, but those final 8 miles were by far the most painful and miserable miles of my life. It took me a good 10-15 minutes just to get out of the finishing area and I was lying in the grass in pain for another 30 minutes trying to rehydrate.
Considering I do have a running background, and the fact that I got in 3 weeks of running beforehand, I'd say it's not a stretch to run a marathon off of minimal training. Just don't try to race the whole thing like I did.
As far as no running whatsoever, I guess it can be done if that person has been active in the past, but it'll have to be at a low intensity.