There is a lot of good input here, but some of it, while good practice, is quite frankly overkill for a lathe in a hobby gunsmithing shop. To the extent overkill keeps a guy (or gal) from getting a lathe and enjoying it, overkill is bad. So I'll offer the following opinion based on my experience.
I understand it's better to have a lathe on a foundation that goes to bedrock with lots of rebar in it, leveled to a gnats eyelash, but frankly, in 60 years of running a lathe on a hobby basis I've not found it necessary for lathes of hobby size - 12 x 36 or even 14 x 40 (or even my friends 15 x 40). I've had my two lathes in three garages and now my detached shop building. Until now they sat on a 4" poured concrete floor in a So. Cal. housing development, earthquakes (Sylmar, Northridge) and all. And it worked just fine. Now it's sitting on a 6" poured concrete floor.
This is especially true of barrel work through the headstock. The carriage travel is two or three inches right near the headstock - if you don't do anything but set a decent lathe on the swept floor and go to work the barrel tenon will be just fine as long as it's correctly aligned with the spindle axis which takes patience but is relatively easy to do with the right tooling.
The tail stock with a home made reamer pusher that has radial float like mine (made in less than two hours) will chamber a barrel through the headstock dead true even if the tail stock is aligned .005" high as long as the bore is correctly aligned with the spindle axis. I know that because that is the case with my 12 x 36 import lathe (sitting on a 6" concrete floor with 2" square x 3/8" thick steel pads under the 4 leveling screws).
Tail stock alignment is really critical only if one doesn't use a reamer pusher that has radial complaince. So use one that has radial compliance. Radial compliance is as easy as the reamer being pushed by a flat surface! The reamer will align itself with the bore unless you push it out of alignment. That fact is completely ignored by most. The reamer is constrained by the physics of it's application to want to exactly center itself in the bore. The only time it won't is if the operator forces it out of alignment or the bore is aligned so that the reamer can't stay centered in it because the pusher won't let it.
Crowning only requires cross slide travel with the carriage or compound moving a few thousandts for creating about any crown one can imagine except Richard Franklin's quiet crown and that one doesn't require precision for the bore. For all the barrel work, the cuts are light, the loads on the lathe are relatively light, and the small movements mean that even if the bed is off .003" from end to end you still won't see it in the tenon threads or the chamber working with the barrel correctly aligned through the headstock.
Even for action truing the loads are light, speeds are reasonabe, and carriage travel is relatively short. It isn't necessary to have a lathe leveled tracable to the national bureau of standards to do it. Use a precision level and leveling screws and you're good to go. I know because that is how I've done it.
Pressure fed coolant is better than stopping, cleaning, dipping the reamer in cutting fluid, and starting atain every 0.015" to 0.025" depth of cut, but millions of chambers have been cut with out pressure fed coolant, and more will be cut every day with out it - probably around the world there are a hundred being cut with out pressure fed coolant as I type this. For production work it would be mandatory because it would make the process so much faster, but for a hobby guy like me, taking 3 hours to cut a chamber means I have 2 hours and 45 minutes more fun per barrel than if I did it in 15 minutes.
Chad Dixon has a wonderful setup, I drool all over pictures of it and read every one of his posts more than once, but one can make very good and accurate rifles with equipment that costs less than 10% of what he has invested. It just takes longer. He can turn them out in quantity but for hobby gunsmithing that isn't required. He is trememdously skilled - if I could afford it I'd just have him build my rifles, but I can't and I love doing it myself. He is also very good at programming the CNC machines, a skill that isn't learned over night.
It isn't necessary to have a professionally reconditioned Monarch or Hardinge either. There are gunsmiths all across the country making their living with Grizzly gunsmithing lathes sitting on 4" thick garage or shop floors. They have happy customers that come back for more. If it works for them, it will work for a hobbiest as well. It works for me.
We aren't trying to make rocket engine parts on a production basis. What's under consideration is a hobby gunsmithing lathe for the original poster, or folks with a similar desire. Barreling, chambering, maybe some action trying, and support for bedding and stock work. Either Grizzly sitting on a 4" or 6" concrete floor (that isn't cracked under the lathe) will do the job - so will a number of other lathes in the same price range. I recommend Grizzly because I've seen them in the show room, seen photos of them being used by professionals, and I think they can provide long term parts support should there be a problem. A lathe is a long term investment. My 9" South Bend was bought by my Dad in 1950 - I still use it several times a month in my hobby shop 60 years later.
The most important thing about a gunsmithing lathe is to have one that's adequate. The perfect lathe you don't have doesn't get it done. What it needs is relatively simple:
1-3/8 minimum thorugh headstock, 1-1/2" is better. 2" is even better but one can make lots of rifles with a 1-1/2" spindle bore.
Maximum of 70 rpm for minimum speed, 1/2 that is better. both my lathes have a slow spindle speed of 36 rpm. I'm about to do the square threads on a 1903 Springfield and I'm thinking of changing the pulley to make it even slower.
Maximum of 18 or 19 inches from the front of what ever is used for a chuck to the centering bolts in the tail of the headstock.
The ability to do both imperial and metric threads.
The tooling (measuring instruments, cutting tools, work holding, tool holding, reamers, thread bits, etc.) to accomplish the work with the lathe.
A reasonably flat uncracked section of concrete floor to set it on far enough from a wall that you can walk behind it.
An extremely desirable option IMO is to have a 3 phase lathe powered off a converter (converters are dirt simple and easy to make). A three phase lathe so powered can be plug reversed (flip the switch from forward to reverse) which is very handy for cutting metric threads to a shoulder that require leaving the half nuts engaged (which is most of them if the lathe has an imperial leadscrew). This is not a good idea if the chuck is threaded onto the spindle, but most new lathes will have a D1-4 or similar spindle nose and with these it works just fine.
As was posted earlier in this thread, either of the Grizzly lathes, and many other 12 x 36 import lathes meet these requirements. Just don't get a Smithy or some other combination machine - you won't like that for gunsmithing at all.
Finally, don't get a lathe that has a headstock so long the barrel won't reach thorugh it and stick out the back end of the spindle an inch or two. That makes aligning the barrel through the headstock a lot more difficult than it if sticks out both ends like it does on my lathe. It also makes using the alignment method in Gordy Gritter's video essentially impossible, at least totally impractical.
Spend the money saved by buying an adequate lathe on a Bridgeport or BP Clone vertical mill. You will find all sorts of things it can do for bedding, cutting extractor notches in barrels, cutting sight dovetails, etc.
Fitch