Production is to begin this fall on the show, which is about the early days of the U.S. space program.
Mark Lafferty -- whose credits include Castle Rock and Halt and Catch Fire -- is to serve as showrunner.
Game of Thrones and Band of Brothers' helmer David Nutter is set to direct and executive produce the first episode.
"The behind-the-scenes stories of the astronauts in Tom Wolfe's bestseller The Right Stuff are engaging, provocative and timeless," Carolyn Bernstein, the channel's executive vice president of global scripted content and documentary films, said in a statement Sunday. "The book's narrative aligns perfectly with the qualities that we look for in scripted projects: fact-based, wildly entertaining and pushing the limits of human achievement."
No casting for the series has been announced yet.
National Geographic and DiCaprio's Appian Way production company previously collaborated on the climate-change documentary Before the Flood.
DiCaprio's acting credits include What's Eating Gilbert Grape, Romeo + Juliet, Titanic, The Aviator, The Departed and The Revenant.
It was announced in October that DiCaprio and director Martin Scorsese are reuniting for a film adaptation of the book Killers of the Flower Moon, a crime drama about the Osage Nation in 1920s Oklahoma.