Watch the video from The Q where he builds a custom shaft driven bicycle.You could start with the front end of a bike, and part of a frame, but the whole wheel, shaft drive, gears, and back of the frame will have to be custom.
source/image(PrtSc): The Q
These bicycles use the drive shaft instead of a chain to transmit power from the pedals to the wheel,this then meshes with another bevel gear mounted on the drive shaft, thus allowing the axis of the drive torque from the pedals to be turned through 90°.
The drive shaft then has another bevel gear near the rear wheel hub which meshes with a bevel gear on the hub where the rear sprocket would be on a conventional bike, and canceling out the first drive torque change of axis.
Shaft drives operate at a very consistent rate of efficiency and performance, without adjustments or maintenance, though their efficiency has been lower than that of a properly adjusted and lubricated chain, possibly because of insufficiently precise machining or alignment of the bevel gears.//wikipedia