I want to fix up the stairs going down to my basement, but I don't want to spend $2000 plus for a new set of stairs with hardwood treads, etc. The current stairs are 'builder's stairs', made of spruce stringers and treads, and MDF risers.

Is it feasible to cover the treads, risers, and stringers with 1/4 inch veneer plywood (say oak or cherry), and glue/nail on a sold wood bullnose, with maybe a lip on the bullnose that extends onto the tread around an inch and which is also 1/4 inch thick so it butts up to the new plywood on the tread?

I will be putting carpet on the treads, leaving about 4 inches of wood exposed on each side, so I don't think wear on the veneer should be a problem.

You can do what you described but do all you can to strengthen the original stairs so they won't squeek with the new veneers.

You may want to look into this new system called "NuStair refacing stair system".