Hkmp5sd has it right.
The shortest length I know of is 6.5 inches.
http://www.serbu.com/shorty.htm
Some of the above answers are incorrect.
They list dimensions for shotguns that are not considered NFA weapons. Shotguns shorter than those dimensions are absolutely positively perfectly legal according to federal law. There isn't a loophole. This isn't a trick question. This isn't some tongue in cheek answer.
The question was "Does anyone know the shortest length of barrel you can have on a shotgun?"
Yes, there is no minimum length according to federal law. Your local and state laws may vary.
A shotgun shorter than the dimensions given above while perfectly legal, require a federal tax stamp which in no way, shape, or form makes it illegal.
I own a shotgun shorter than dimensions given above and am not a criminal. In fact, I have a piece of paper sent to me by an agent of the federal government giving me permission to own it. I am getting ready to buy another one (the one linked in this post) and it is also perfectly legal. If I am wrong, then I must be a criminal, right ?
In my opinion, the firearms laws in this country are far too strict. I personally believe that we should not have laws limiting our right to own, or use any firearm we desire: but that isn't the case. We do have laws limiting our rights. What we don't need to do is make up laws that are more strict than those already on the books. There is no federal law saying we can't have a shotgun barrel shorter than 18", so why make one up ? In fact, the exact opposite is true. Federal law specifically says we CAN own shotguns with barrels less than 18" and shorter than 26" OAL: and they tell us exactly what is required to own such a shotgun. I enjoy and exercise my rights as an American to buy legal firearms.