What kind of lathe do guys use for this stuff. would you be able to get away with something like the one in the link, or would you need something a lot longer?
https://www.homedepot.com/p/Grizzly-Industrial-8-in-x-16-in-Lathe-with-Milling-Head-G0769/310410440
That really is too small IMO. You can get a much larger lathe for a lot less than that on the used market anyway, it just takes some searching. I had a 12x39" Grizzly for a while that I bought for $1500, then sold that and have a 13"x36" Jet that I bought for $1650. Both of those machines are much more capable for barrel work than that little 8x16.
You can do some decent work with a small lathe, but it's a lot harder and you have to figure out workarounds. On that particular 8x16 it looks like the spindle bore is small, and the bed is so short that you wouldn't be able to work on barrels longer than about 12" or so.
A big thing to look for is a large spindle bore; for working on rifle barrels I suggest a minimum spindle bore of 1.5". That lets you put a barreled action through the back end of the the spindle to work on one end. For contouring, you need enough bed length to cover the whole barrel plus extra for the chuck at one end and live center at the other. Depends on the tooling but I'd say at least 28-32" for most rifle barrels, and longer is better.
You'll also find out that barrel profiling is not an easy task on a general purpose lathe like that. It can be done, but you have to deal with chatter, and come up with ways to handle tapered profiles.
