What a cool idea! The real limiting factor is that you could only ever really sleep for like 15 mins at a time, maybe 20 mins if you fall asleep almost immediately and get up 5 mins before getting going again.
So how often can someone last on sleeping 15-20 mins an hour? I have no idea - if you did it for the full 24 hours, you would have gotten 6-8 hours, but it would definitely not be good quality and you wouldn't have ever touched the REM phase.
Then, on top of that, you've got to complete the run... man, this is tough to call. I'm guessing this would be incredibly difficult to maintain belong 48 hours, but I would bet the race would be over long before then.
I remember a great thread from years ago by Ian Edwards, who ran 2 miles around his neighbourhood, every hour, for 24 hours. He struggled even with half the distance, and with the psychological boost of having a defined end point to shoot for - but then, he wasn't an unltramarathoner.