When “From the Earth to the Moon” premiered in April 1998, it was a big deal for HBO. Still two months away from debuting “Sex and the City” and eight months out from “The Sopranos,” what was then still known as the Home Box Office network was making a big prestige play, with a miniseries about Nasa’s Apollo missions, in a genre dominated by broadcast. After all, broadcast networks reached more people, and only cable providers willing to pay a little extra could check out HBO’s biggest event to date.

Of course, that made it a pretty big deal for TV itself, as the shows dominating cultural conversations and the awards circuit started to shift away from the Big Four networks and into the cable space. There was no way of knowing this 21 years ago, but “From the Earth to the Moon” was an event nonetheless — costing $65 million, shooting on more than 100 locations,...