Interesting....
Like JonB said, it would sit very, very low, which in and of itself wouldn't be so bad. The issue I see right off as being a deal breaker is reaching the pedals. Because you are reaching around the fork with your legs, you would have to be extremely bow-legged, or increase the Q factor with some pedal extenders. Even then, I think you would have to have an extremely high bottom bracket to reach. I'm not saying it couldn't be done, I'm saying it might not be practical.
I don't think the lack of suspension would kill you because it looks to be a pretty long wheelbase. Maybe too long, hence the need for a high bottom bracket to reach the pedals and or clear the fork crown. Now, all that being said; If you could somehow weld a cross bar between the seat tube/seat stay juntion and the downtube.... Then you might have something! That might be really cool! Then you would be well above the fork crown with your legs and could probably add an aftermarket suspension fork, if you saw the need. See the attached for an example of what I am talking about.
Mark