Not a film, but 'Apollo: Race to the Moon' by Murray and Cox is a book covering the engineering and management efforts that went into Apollo, and is the closest

Not a film, but 'Apollo: Race to the Moon' by Murray and Cox is a book covering the engineering and management efforts that went into Apollo, and is the closest, most interesting and entertaining thing I've found that is similar in spirit to the LEM episode of 'From the Earth to the Moon.'