I could be completely full of it, but earning my living as a welder for the last 30 years, I suspect that any carbon stell worthy of being a barrel would need to be heat treated after welding. 4140, a very common variety of steel used in gunmaking will develop cracks just on cooling down without treatment. Taking any 'ol barrel and welding on it is risky on a good day.
I still think the best way to get a barrel on a Handi is to sacrifice an existing barrel., cutting it off just forward of the lug and reaming \and threading it for another barrel. This is a good method, just takes a competent machinist. For most of us, time we buy a barrel blank and have a gunsmith/machinist turn it into something useable, we'd ahve the best part of another rifle invested.
A reboring joib may be the best alternative, start with a 223 bull barrel and work from there. Fred has shown the way in this.