The Internet Archive has a trove of documentaries about NASA's Apollo Program, produced by NASA in the late 1960s and early 1970s. The blog Silver in a Haystack is dedicated to finding this kind of thing—great, free films available online from the Archive. A year ago, the blog highlighted this 12-part series of Apollo documentaries (including download links to those available online). The author of Silver in a Haystack wrote:
The editing and narration are fairly low-key. There is some focus on the technical aspects of the missions, and compared to many later Apollo documentaries, there is little about the astronauts’ feelings and practically nothing about their private lives. There is also nothing about the politics behind the missions. For my own part, I find this to be a refreshing change. We get to focus on the process of getting men on the moon, and that is as it should be.
While I haven't watched all of them (they're about a half hour each), I jumped on two cultural touchstones, embedded below.
Apollo 13 - Houston, We've Got a Problem
While the Hollywood Apollo 13 movie is excellent, this is a terrific documentary showing the real people and the real moments involved in the Apollo 13, er, problem. It's great watching real footage of the men and women of NASA working the problem, chain-smoking, and generally struggling to get the crew home. We also get to see citizens (in church services and at baseball games) praying for the return of the crew. Seeing the public reaction when they make it home? Truly beautiful.
Apollo 11 - The Eagle Has Landed
The audio is a bit low, and the smooth narration might ease you into a nap, but it's a straightforward look at the Apollo 11 mission. This is the kind of thing we used to watch on filmstrips in school.