For a $2K budget, you are looking at Ebay, Kijiji or Craigslist, until you luck in to a decent used lathe for cheap. With any luck, you will get at least a basic set of tools with, stuff like steady and follow rests, a tool post of some sort, and like as not, a few basic tools. Expect it to need a thorough cleaning. Most of the lathes that sell new for your budget, are not nearly large enough to accomplish what you want. Figure maybe triple what you have budgeted, if you want new, maybe more. Potentially, LOTS more.
You can do accurate work on any lathe, once you understand what the lathe needs from the operator to make it happen. Like as not, there will be a pretty solid learning curve to get past.
Lathes are cheaper to tool up than a mill is, and I doubt very much that you wold 'need' a thousand or more dollars in tooling to accomplish what you want. You could spend that and more and still not have what you actually need, so shop wisely.
Got any experience running a lathe? If not, then you are in for an experience! If you manage to stick through the learning curve without tearing your hair out, and giving it all up for a less stressful hobby, you might soon enough figure that your goal of doing barrels will need a few skillsets that don't exactly come from books very well. That is to say, the hands, eyes, and brain of the operator, are where the magic happens.
In that regard, I would suggest that ANY lathe is better than no lathe, and you can learn a lot about cutting metal on a lathe that may not be big enough to do the end goal work. Making some of the toling that you might be inclined to buy, is a good use of your time if you do not already have the lathe skills. Raw stock is cheap, compared to even used barrel blanks. Making stuff like floating reamer holders or a flush through system for chambering, can allow you to learn the skills while still accumulating useful items.
Cheers
Trev
