Going by the size of your lathe (15" x 5') you can make a barrel four feet long or maybe 4'4" without too much trouble, or maybe lots of trouble. The steel I would use would be SAE 1018 or SAE 1025 or a steel within this range. Refer to Machinery's Handbook Edition 24, page 381 for details on these steels. The first operation is to drill the hole rather than profile the barrel. If you profile first and then mess up the hole beyond reclamation, you will have completed hours of work for great loss. Decide what caliber you want and then decide what degree of finish you want in the bore. A fairly good finish can be formed by a standard twist drill and then reaming with a machine reamer.
These tools will need to be extended in length and are held in a fixture in the toolpost, not in the tailstock. A high pressure pump (250psi to 300psi) is needed to cool the drill and more importantly, to flush out the chips. I made a standard gun drill when I drilled out my own cannon. The coolant is deliverd to the very end of the drill with this style of gun drill. These drills produce small chips that will flow out with the coolant. A drill that produces "ribbons" is harder to use as the ribbon swarf does not lend itself to easy removal from a deep hole. All you need is patience and time to do the opperation.
I strongly recomend that you start the hole by boring the hole to the exact size of the drill as deeply as possible. This will start the drill exactly in the center and provide a perfect guided start for the drill to continue down the shaft. A pilot hole can be run in to help take the pressure off the finishing drill but only when the initial bole has been bored out. Don't run the pilot hole deeper that a couple of inches at a time. A 1/2" drill would make a good pilot drill. This pilot drill MUST be sharpened perfectly true and correct. If the pilot runs off, so will the finishing drill. This is not good for a reamer when it tries to straighten the hole to finished size.
The barrel is held at the end by a three point steady that seats on a machined diameter to ensure proper accuracy when drilling the bore. The hardest part is getting the swarf out of the hole. If the bore is bigger than the drills and reamers you can afford to buy then you will need to make a gun drill to suit. If you decide to go ahead with the project, contact me and I will help you with the design and construction info for the drills.
Joe (Wedge)