In the 30's and before, some cooks and domestics in the South used to be known to carry a "broken arm." They would take wrapped food home in their arm which would be held close across their chest like a broken arm.
Definitions include: A slang way of saying too much money. Compare to: expensive, High Priced or costly. It's going to cost me an [Arm and a Leg] to buy that Bentley my girl wants. I guess she wont be getting it..