Hi Ian, below are a few photos of my Myford tailstock barrel, the first one shows the front end with the No. 1 Morse taper and the keyway on the side of the barrel to stop rotation. Just to the left can be seen the pin through the body, which has a key cut on the inside end which will prevent the barrel from coming out of the front end. this pin doesn't seem to be easy to remove, but yours may have a screw in it as Bazyle suggests, if it doesn't have a screw, you will have to take some drastic action and cut the chuck off to enable you to remove the barrel from the back end.

To remove it from the back end you will have to remove the two screws that hold the keep plates in position.

These should be accessible through a hole in the handle.

Once they are removed, the barrel and handle will be able to be removed from the back end.

Once removed you should be able to deal with the barrel, but of course you might need another lathe or take it to someone who can do it for you, or find another barrel from somewhere.
I'm beginning to think this might be what I call a Johnny Cash job, i.e. his song "One piece at a time" where he nicked a whole car from the factory he worked in and had problems with changes in designs over the years. Maybe this lathe was got by the same means, but the guy got the boot before he got all the right bits and the right colour paint. 
Regards Nick.
P.S. I'm not saying that Johnny Cash actually nicked a car from a factory he worked in, but you get the idea from his song.
Edited By Nicholas Farr on 07/01/2019 10:09:06