How to repair an old wooden barrel?

I have that same issue!

try some exterior wood clear epoxy

Get some wire and wrap around the barrel and twist really tight. You can Go back and glue some roping to cover the wire. You may need to do this in a couple places around the barrel. You will need help in doing this, extra hands please.

iIs the problem the banding has dropped done and become loose or has the banding broken? If it is loose get a strapping clamp, that is like a nylon tie down strap without hooks, place this above the metal straps old location and use it to pull the wood slats tightly together. Then pull the strap up as tight as possible and then use wood blocks to drive it up further evenly. Use it's old location as a guide. One could use heat to expand the metal strap and once driven to a tight position it is cooled with water or oil to make it retract and tighten further, I think that is how barrel makers do this. If the strap is broken and is only rotted or corroded in one spot you can try to bend up about 3/8" on each end and drill a hole through these and then put a small bolt through these and as a nut is tightened on this it will pull the banding together tightly. You could make banding but this would require a heavy gauge strap be angled by hammering the top edge to lengthen it. You could also use thin wood strapping but that would need to be steamed and nailed in place. If looks don't matter get somebody at a lumber yard to put a metal or plastic strap, used for banding shipments of lumber, around it. You can also get large hose clamps, a foot or more in diameter, and join them end to end but they have a limited torque setting and break if overtightened.