Picard takes to the holodeck for a little R&R, playing the role of Dixon Hill, a 1940s private detective that would be at home in a film noir, if only this episode had the gumption to actually do noir as a style rather than simply a generic concept. If the point of this episode is merely to do a period piece with 1940s costumes and sets, it's a success. If the point is to tell a compelling story, it's a failure.
The funny thing about TNG season one is its pace; at times it's almost startlingly slow, with simple, straightforward plots. "The Big Goodbye" is an example of not just slow, but also uneventful — far too much so for its own good. There simply isn't an hour's worth of material here, and the payoff is too lacking in juice and irony to be worth the wait.
It's the first Holodeck Run Awry episode — a TNG concept that would go on to become a tradition and ultimately a cliche. I should probably note, however, that "awry" is far too extreme a word for this exceptionally restrained episode. Even tough-guy actor Lawrence Tierney, as big gangster Cyrus Redblock, seems hobbled by the episode's restraint. His right-hand man Felix (Harvey Jason) is more colorful, but also far more annoying, and way too stupid to be plausible.
There is one interesting question that the episode poses: Picard exits the holodeck and leaves Hill's cop friend pondering whether his life is simply an illusion — which, of course, it is. It's a question that would surface many more times in Trek after this story.