Wordorigins.org

The phrase over a barrel, meaning to be helpless or in a dire predicament, has been in use since at least 1939 when it was used in Raymond Chandler’s The Big Sleep:

We keep a file on unidentified bullets nowadays. Some day you might use that gun again. Then you’d be over a barrel.

The metaphor is probably a reference to a prisoner being strapped over a barrel and flogged. Literal references to a barrel being used for flogging date back to the 19th century. This poem from 1869’s Nonsense by Brick Pomeroy uses over a barrel to refer to children being punished by a schoolteacher:

I’d like to be a school-marm,
And with the school-marms stand,
With a bad boy over a barrel
And with a spanker in my hand