Channing Tatum and Scarlett Johansson star in “Fly Me to the Moon.” Dan McFadden/Sony Pictures Entertainment

What do you get when you cross “Apollo 13” with a Rock Hudson/Doris Day romantic comedy? Wait, why would you even try? “Fly Me to the Moon” is a platypus of a movie, and while it definitely has its charms, bizarro and otherwise, you may be more fascinated by the baling wire holding it together than what’s actually in it.

Still, Scarlett Johansson and Channing Tatum at odds and in love – how can that not be a good deal in the on-screen chemistry department? And it is, even if the two stars have to swim upriver against contrived dialogue and some unflattering costume and hairstyling choices. The year is 1969, and NASA is fighting a rearguard action to launch the Apollo 11 spaceflight in an America exhausted by war, assassinations and protest. Why spend money to send men to the moon when everything on Earth is about to collapse?

Harried mission director Cole Davis (Tatum) rouses his team to press ahead, but someone high up the food chain in Washington decides a PR campaign is in order and dispatches a gleefully sinister government spook (Woody Harrelson) to hire Kelly Jones (Johansson), a Madison Avenue ad whiz first seen confounding a conference room full of piggish Ford executives. Kelly is flown down to Cocoa Beach, Florida, with her sassy young assistant (comedian/actress Anna Garcia) and sets about selling the moon landing to America while driving Cole crazy.

As long as “Fly Me to the Moon” stays on that simple level of Irresistible Force (Kelly) meets Immovable Object (Cole), the movie is decent fluffy fun. Kelly lines up product sponsorships (most of them real – remember Tang?) and gives astronauts Neil Armstrong (Nick Dillenburg), Buzz Aldrin (Colin Woodell) and Michael Collins (Christian Zuber) an all-American-hero makeover. She humanizes the mission, in other words, and provides it with leading men and a storyline. All of which gives Cole – a fighter pilot and a scientist congenitally allergic to lying – the hives.

Tatum sells his character’s square-jawed sincerity with an undercurrent of old-school patriotism, and he’s good enough to make you forget how sharp his comic timing can be. “Fly Me to the Moon” is built on classic screwball-comedy bones, where the man represents order and the woman is pure chaos agent. Think Hepburn and Grant in “Bringing Up Baby” – Mr. Orthodoxy, meet Miss Catastrophe. But while Johansson does her best, she may be too thoughtful an actress for Kelly to really let loose. There’s no madness in her, and so the romance never fully takes flight.

The screenplay by Rose Gilroy, from a story by Keenan Flynn and Bill Kirstein, doesn’t help, since it’s juggling the love story with a serious treatment of the Apollo 11 mission preparations, and that’s before the sinister government agent reappears to insist Kelly produce a hush-hush fake moon landing on a soundstage, in case the real one goes south. (I’m not telling you anything that’s not in the trailers.)

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True, one of the lovers in a rom-com always has to have a secret for there to be a second-act betrayal and argument, but this – this is nuts. The film’s staged moon landing, of course, is a nod to conspiracy theories that arose in the 1970s positing that the Apollo program was a NASA hoax directed by Stanley Kubrick (who gets a passing shout-out in “Fly Me to the Moon”) and designed to hoodwink the Soviets. (1977’s “Capricorn One” helped popularize the myth, too, substituting Mars for the moon.)

Director Greg Berlanti (“Love, Simon”) gets some healthy slapstick out of the soundstage sequences, but the whole subplot has a queasy feeling, as if even pretending the hoax really happened is enough to reawaken the crazies. But all of “Fly Me to the Moon” is like that: broad farce, serious historical drama (there’s more than one visit to the memorial for the fallen astronauts of Apollo 1), light romantic comedy, suspense and more, all wrapped up in two hours and 12 minutes of lumpy entertainment. (Resolved: A good rom-com should last no longer than 90 minutes.)

Some fine actors get lost in the clutter – Garcia, Ray Romano as Cole’s supervisor, Noah Robbins and Donald Elise Watkins as junior NASA employees. As the screamingly gay ad director hired to film the fake landings – he’s named Lance, for pity’s sake – Jim Rash is swish enough to make you miss the subtle comic stylings of Paul Lynde. “Fly Me to the Moon” strains to achieve liftoff, sometimes quite amusingly. But in the end, it’s just too heavy to get off the ground.

Ty Burr is the author of the movie recommendation newsletter Ty Burr’s Watch List at tyburrswatchlist.com.

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