The barrel would almost certainly have to be metal. It might be possible to make a barrel out of some exotic ceramic material since we have knives now with ceramic blades but a barrel has to also withstand quite a bit of pressure.
Steel barrel due to incredibly high pressure. Aluminum would burst. Ceramics will fracture and explode.
Winchester Model 59 shotgun, circa 1960:
The Winchester Model 59 features technological advancements and innovations never seen before when it was released in 1960. A six-pound twelve gauge semi-automatic that is reliable and easy to clean proved it to be ahead of its time. There are no gas valves, inertia systems or complex mechanisms that make this gun function. A super-simple floating breach is the heart of the gun.
The floating breach consists of the chamber and forcing cone, however it is a separate piece from the barrel. Upon firing, the floating breach acts like a short piston and moves to the rear about 1/10 of an inch. That slight movement sets the bolt into rearward travel under its own momentum. A return spring pushes the bolt forward again where it picks up and feeds another cartridge.
The innovation does not stop there. The barrel is ultra light-weight. It is a metal tube that is wrapped in over 500 miles of fiber thread that fused and bonded. This process is similar to the carbon-fiber barrels that are increasingly popular on rifles.
It might also be possible to use a ceramic/glass composite inner *base tube* rather than metal. Composites and ceramics have come a mighty long way since 1960. Remember that early muzzleloading rifles used Damascus barrels made by forge-welding three bars of metal radially around a mandrel.