My (annoyingly) omnitalented friend Eric Felten sends this along:

Hi Jonah,
Hope you’re doing well.
The expression was not widespread until the 1970s (interestingly, because perhaps the average person didn’t come into contact with lots of government work until then).
But there are two competing claims for the expression worth considering. The first, according to “Projection of War: Hollywood, American Culture, and World War II” says that the phrase originated in Hollywood: “The ironic catchphrase that studio system personnel coined during the Second World War — good enough for government work — slighted not the significance of the work but the competence of the government to evaluate it.”
But I think the more likely origin can be found at the Smithsonian Institution. According to a biography “Smithsonian Institution Secretary Charles Doolittle Walcott” written by an accomplished scholar, Ellis Leon Yochelson, the original phrase is “close enough for government work.” The Smithsonian had put out a large document on purchasing which included three pages specifying how display cases and drawers were to be built. According to Yochelson, the instructions detailed “the kind of wood, its thickness, number of nails, and so forth, and ended with ‘or close enough for government work.’ ” Unfortunately, Yochelson isn’t clear on exactly when the document was written.
Hope that helps.
cheers,
Eric