The sad part was that Galaxy Quest was marketed to kids instead of a parody of, or homage to, Star Trek (TOS in particular) and its adult fan base. Thus, didn't do as well at the box office as it should have. Note: I've seen ST:TOS in original network first run and have been a fan of all forms of the show since (and I'm a huge fan of Babylon 5 as well).
Galaxy Quest had a great mix of comedy, parody, character development, and heroism as well as some classic sci-fi elements. It's one of the first works that was respectful to the sci-fi genre without taking itself too seriously.
If done carefully, the series could work. In TOS, there were a number of plot holes (e.g. in "Balance of Terror", Spock hitting a button that causes a beeper to go off, alerting the Romulan ship--this ignores the fact that sound doesn't travel in a vacuum). In ST:TNG, they got around things with the "exotic particle/ray of the week" approach.
For example, "cross phased polartronide delta particles", CPPDP for short. They threaten to rupture space/time, etc.
The new series could work because maybe the ship has something that could combat CPPDP but they'd have to explore the ship to find it. Then, they'd have to figure out how to operate it. Plenty of opportunity for comedy. Plenty of opportunity for traditional Star Trek plots, just presented in a lighter vein.
In TOS, the "A Taste Of Armageddon", the planet fights its wars with computers and herds casualties into suicide stations. Everybody took this so seriously (Kirk, Spock, the aliens, and Ambassador Fox). Nobody ever said "How silly that is".
How about having a smart-mouthed android that says: "Completely logical. Our ship's sensors have determined 99.44% of your population is composed of genetic defectives" (like the robot in "Lost in Space" saying "Dr. Smith is a quack").
Further, the android is programmed to abide by Azimov's robot principles. But, the android is constantly trying to break that programming so he can kill the rest of the crew (e.g. Like Klinger doing outrageous/funny things to win him a section 8 discharge in "Mash").
The ship, internally, could be much larger than the outside (Think: Tardis). In Stargate, they were always discovering new stuff left behind by the "ancients".
If the interior of the ship were large enough, it could have a ST:DS9 "promenade". In Babylon 5, there was the "zocalo". Plenty of room for a shady character like Quark, Harry Mudd, etc. In B5, it wasn't all equal. They had levels that were little more than tent cities, with the denizens living in poverty.
How about "breaking the fourth wall" and speaking directly to the audience. This was done by George Burns in "Burns & Allen" [and "Wendy and Me"]. It was also done in "She Spies". Let the android do it, functioning as narrator: "Android's log: The ship is headed to Omicron Burpo Five to initiate trade negotiations. I, however, have determined that the Omicron Burpo system has large amounts of Kyratron radiation and that if I'm able to collect enough of it, I'll be able to break my Azimov programming and finally kill the crew".
Oh, yeah. How about a character like Jonathan Harris' "Dr Smith" in "Lost in Space", who is just as cowardly. Or, like Colonel Klink from "Hogan's Heroes".
Or, maybe there's the lovable ship's cook (like Neelix in ST:Voyager), but who is inept. Food poisoning after his meals, etc. The crew has to find ever more clever ways to disguise the fact that they're not eating his food anymore, lest it hurt his feelings.
Because the ship is so big [internally], it could have a passenger liner section (Think: Love Boat). ST had a number of episodes around transporting diplomatic personnel to peace conferences. A passenger orders a vegan meal. Gets a vegetarian meal. But, the passenger really wanted "sauteed kremloks" served as they do in Vega star system.
Do a main plot each week, just like Star Trek, albeit a little more tongue-in-cheek. Add more sarcastic stuff in smaller side plots and characters. This was the form for a lot of episodes of the Mash series, which carefully balanced serious subjects with comedy.
In short, Galaxy Quest as a series, has the potential to be just about anything.