Zoom.......
To cut through some of the possible confusion.....
There are, indeed, many makes of lathe which would do your job.....but there is only one which I would recommend to you, for practical serviceability, cost-effectiveness, and "operator-friendliness" for an inexperienced operator.
That one is the older South Bend, in the 13" or 16" size, and only in the later "toolroom" version, which was supplied with the hardened ways, D1-4 camlock spindle nose, large, easily read dials, 5C collet closer, and the taper-attachment.
The South Bend uses bronze sleeve bearings, which are provided with a sensitive adjustment, so the South Bend spindle runs very smoothly, and so will do very good workpiece finish.
The South Bend, of course, can never take the place of a heavy gear-head lathe for roughing out large steel parts, but, if you don't anticipate any heavy production work, that's not an issue.
The South Bend, with its sleeve bearings, is limited to an approximately 900-ish top spindle speed.....which is not a problem at all, for the generality of non-production fine work.
Fact is, for one-off parts, and such as rifle barrel work, you'll be running slowly and carefully anyway. Hogging the metal away rapidly with carboloy tooling at its optimal speeds is simply not likely to be part of your work.
One major advantage of the South Bend is its simplicity and ease of maintenence, and the ready availability of South Bend parts and tooling items, should they be needed.
Unfortunately, you may need to invest some time in looking for a nice late pattern South Bend.....there are many of the earlier types around, which can be bought for almost nothing, but they are the thread-spindle, soft-ways version, which will often be found with significant wear in the ways, near the headstock....if you can find an unworn one, and the price is right, you might "get lucky", but be careful, and get someone to show you how to inspect a lathe for wear....its not cost-effective to rebuild an older South Bend, considering the cost of re-grinding the ways, etc.
cheers
Carla