Spoiler Level: High
This episode gets a bad wrap. Yeah, it has some bad moments, but all in all it's a fun little episode.
The Enterprise comes across the remains of an old NASA expedition, much farther out than any such expedition could have reached. On the planet's surface, Riker, Data and Worf discover a great big nothing with a pair of revolving doors. Entering leads them into a bizarre recreation of an old Las Vegas hotel. It's not long before they discover that the entire thing was made by aliens as a place for the last surviving astronaut from the NASA mission, based on a cheap paperback novel found on the NASA ship. But just like the Hotel California, at the Hotel Royale you can check out anytime you like, but you can never leave.
There's very little I really don't like about the episode, but one of the things I do dislike is that there was no good way at the time to do an effect of the Away Team going through the revolving door and coming out into the same room (à la the most recent Doctor Who short, "Space"). It just makes it look like Riker, Data and Worf are the Three Stooges and aren't smart enough to know how to use it.
The story itself is a cool concept, and it's always fun to see characters out of their element. The scene where a phone rings and Worf isn't quite sure what to do with it is great, and Data's gambling at the end is priceless. And I always liked the part where they identify the timeframe of the mission based on the fact that the US flag has 52 stars. (Wonder if that will ever happen? We've got 22 years left before this becomes inconsistent.)
Speaking of inconsistencies, I never would have thought to look this up if it wasn't (once again) for Larry Nemecek's The Star Trek: The Next Generation Companion, but this episode uses Fermat's Last Theorem as a parallel for the NASA mission; while they may have learned what happened to them, they still don't know why it happened or how they got to be so far out into space. Picard muses that, like Fermat's Last Theorem, it's a puzzle that may never be solved. (Which kind of bugs me; it would have been just as easy to write an answer in the journal they found. It stikes me as attempt to be mysterious that doesn't really work.) As of the book's printing (at least my copy, which is the May '95 revision), Nemecek says that as of 1993 Fermat's Last Theorem was close to being proven, and a quick Google search shows me that it was in fact solved... also in May of 1995!