In the 60 years since the first episode of “The Twilight Zone” aired, other television series have lasted longer, attracted more viewers and left a deeper footprint in popular culture. When we want to suggest that a certain politician is logical and collected, say, we call him Mr. Spock, and when we want to poke fun at self-dramatizing cool, we still turn to the Fonz. “The Twilight Zone” may not have the same reach, but it continues to play a special role in the popular imagination—when neighbors and friends are torn apart by politics, when technology goes too far too fast. That’s why the show, whose original run ended...