Good bits and pieces to have lying around the house if you want to build a jet engine would be a car turbo. How small do they make those?
I would probably go with a design similar to the Sundstrand APS1000 APU - weld the (centrifugal) compressor and (ditto) turbine together, lead the air past the combustion chamber lining for cooling (add cooling air holes) and let it enter the flame tube at the far end of the combustion chamber as it turns around back to the turbine.
Compact and not a long shaft to complicate things like lubrication.
Cheers,
/ft