Seems technically possible to me,
but there are difficult issues.
The Arduino simply uses these sensors (say 5 sensors) to know how far away the floor and walls are. If it is too close to the left wall, it gives a little right roll. The amount of right roll would need to be tuned. Probably a PID.
The real difficulty comes say in a hall when the quad is diagonal and does not get a good distance reading to the walls (so maybe you need more like 9 sensors, 8 for the walls). Also, having that many sensors running at once, they will interfere with each other.
You could have the quad rotate, and use less sensors. I recall there is actually a toy copter that does this with one horizontal sensor and one or two vertical sensors. It spins rapidly and won't hit a wall.
I think this is a good task for vision, but that needs a really big processor, like quad core 2ghz; and lots of scientific code.