To turn the legs and stretchers, I had to haul my old Craftsman lathe out of mothballs. I think I paid $75 for this lathe, about 15 years ago, and I haven't used it in nearly that long. After mounting it on a shopmade stand, and hooking up the ancient motor, I was thrilled to find that it still works! Ah, they don't make 'em like they used to. The next task was to make a proper toolrest. The stock toolrest on this, or really any lathe, is about 12 inches long--not nearly big enough for chair work. I got some good ideas from blog posts by Caleb James and Tim Manney, then designed my own version.
As the pictures below show, I made two riser blocks that straddle the tube. I installed 1/4-20 threaded inserts in each, then screwed on a straight, flat 2 x 4 that is slotted (so it can be adjusted closer/farther from the turning axis). Last, I bolted a piece of 1 1/2 steel angle on top. The tool rest on the edge of the angle. This is a very simple rest, and is not height-adjustable, but it works great for the task at hand. I found that it helped a lot to file the edge of the steel angle flat and smooth, so that the tool does not catch as it moves back and forth.