When a film includes a conspiracy theory about the Apollo 11 Moon landing being staged, one might not expect it to get a thumbs-up from NASA, but according to Greg Berlanti, director of “Fly Me to the Moon,” NASA even shared real-life footage from the Apollo program for the film.

I can’t help but think this speaks to how gravity-defying “Fly Me to the Moon” manages to be, in entertaining a long-held paranoid fantasy about the space program, at the same time that its story wholeheartedly honors the actual achievements of all the men and women who worked to get us to the Moon.

In this comically fictionalized version of history, a government spook (Woody Harrelson) blackmails a lifelong scam artist who’s become a legitimate advertising executive (Scarlett Johansson) into marketing an unpopular space program to the American public, right on time for our country’s astronauts to make it to the Moon before the 1960s end.

Because NASA personnel — such as stoic launch director Cole Davis (Channing Tatum) — initially prove camera-shy, our shameless and savvy ad lady simply recruits actors to play them in their first round of TV news appearances.

This foreshadows the staged moon landing that the manipulative G-man eventually pitches as a backup plan, in case the actual moon landing can’t happen or goes wrong somehow.

What’s impressive is how many facts “Fly Me to the Moon” includes among its amusing embellishments.

Not only were the real-life NASA engineers every bit as young as their onscreen counterparts — practically kids in their 20s who regarded 30 as “old” — but the ads in which the astronauts appear in this film were ones that NASA really did sign off on.

(There’s a reason even my generation still remembers Tang.)

Johansson’s character is loosely inspired by Julian Scheer, a journalist-turned-PR man who worked for NASA during the 1960s, coordinating news coverage of Apollo 11, and while Tatum’s character of “Cole Davis” is an amalgam of a number of NASA flight directors, it was Deke Slayton in particular who occasionally butted heads with Scheer.

But no, unlike Johansson and Tatum’s characters, Scheer and Slayton’s antagonism never gave way to a romance, because that would have been way more wild to make a movie about.

I’ll never pretend to be anything but a sucker for films that celebrate NASA’s awesome legacy, and “Fly Me to the Moon” is no exception.

Even when Berlanti lifts a page from Steven Spielberg in “Jurassic Park,” by showing us the sheer physical power of those rocket launches through the ripples and rattling of desktop drinks, it’s still effective.

Beyond the verisimilitude afforded by NASA allowing the film access to its facilities and its archival footage, “Fly Me to the Moon” benefits tremendously from the unambitious but deft performances of its leading man and lady.

Neither Johansson nor Tatum are aiming for any acclaim in their acting, but their respective onscreen personae feel pleasantly unforced.

Tatum plays his stubborn NASA veteran as a sort of lowercase-letters Captain America, a studious nerd who’s blithely unaware of his own model jawline or jock physique, while Johansson simply seems relieved to lend some spunk and pathos to an especially quirky meet-cute comedy.

A scene in which Tatum’s character makes the case that scientific discovery can complement religious faith, rather than detracting from it, might have come across as hokey in another context, but Tatum sells it well.

This feels especially resonant given the divisiveness of recent real-world events, enough that the idea of finding common ground on such issues has become refreshing again.

On the other end of the dramatic spectrum, Jim Rash storms onto set with all the panache that fans of the TV show “Community” have come to expect from Greendale College’s Dean Craig Pelton.

Rash plays the director of the staged Moon landing with a broader artiste streak (but admittedly far less attention to detail) than Stanley Kubrick.

“Fly Me to the Moon” is hardly Ron Howard’s “Apollo 13,” but it’s an endearing little joy-ride of nostalgia, back to when the stars seemed to possess a virtually limitless potential.