The story is File and Forget, and I will say no more about it, since much of its fun is how the thing develops. The cast of the show, most of whom are in this sketch, are gems. Many of the names aren't well-known today, though some are and all were high quality names at the time. Tom Ewell plays Thurber in the sketch, and others include Paul Ford, Peggy Cass, John McGiver and Alice Ghostley. And there's near-perfect underscoring music to complement it all.
One fun historic note. Fun, but one of those stories that makes me SO wish I could have seen it live, or at least if only there was video of it. The story is that James Thurber was a bit of a ham, and for a few weeks into the run, the producers had the brilliant idea to bring in Thurber to play himself! Because he was legally blind, they built a sort of conveyor belt with a chair on it, to bring him onstage and off. And, of my, do I wish there was a video of that.
Happily, there's audio of the selection.