I have made quite a few tailstock barrels, some up to 5″ diameter and 15″ long.
The best way I have found and this is the way that suits me, there are bound to be other ways, is to start off with a piece of over size bar, not much, for a Myford sized machine, 50 thou or so.
First job is to put the taper in the end with it running as true as possible but a 3 jaw and steady is fine, remember you have material to take off. Completely finish the taper including the ejection slot if needed.
Next job is to make a ring gauge that is a nice fit on an unworn part of the original barrel, only needs to be 1″ or so long and polish internally to fit.
Next job is the chuck a piece of scrapbinium [TM] up and turn a morse taper to match the taper in the new barrel, DO NOT REMOVE FROM CHUCK.
then carefully push the new barrel onto the taper in the chuck and at low speed 100 to 150 revs carefully put a centre drill hole in the other end.
You need a low speed to stop it whipping and a completely flat surface to stop the centre drill wobbling on a pip.
Once you have the revolting centre in the end it’s all plain sailing, aim for a thou over and polish to fit starting at the tailstock end and working up in bands trying the ring gauge on for fit.
Do not let the part warm up when polishing, take your time or use coolant to keep it cool.
Also make sure the barrel is clean and lightly oiled when trying the gauge, it’s most embarrassing to find the gauge picks up and the nice new barrel [ don’t ask ]
Rest of the job, keep slot, thru hole etc is simple from here and depends on design.
As an added bonus if you can find a company that does Tuftriding get the new barrel done. Tuftriding puts a very thin hard case on the part but because it’s done at low temperatures, like 450 degrees C there is no distortion and it protects it against the dings of everyday working.
John S.