"Earth ship Ark, man's greatest and final achievement: out of control drifting through deep space over 800 years into the far future, its passengers descendants of the last survivors of the dead planet Earth, locked in separate worlds their destination long forgotten - heading for destruction unless three young people can save the Starlost."
This show was just a dim memory from my youth. I didn't remember its name and had conflated it with the movie "Silent Running" to which I thought it was sequel - until I finally saw that movie to its end. The spaceship model used for the exterior shots was the same, but the story was completely different; one conceived by Harlan Ellison, and also not unlike Robert Heinlein's novella "Orphans of the Sky."
It featured "2001: A Space Odyssey" star Keir Dullea. Walter Koenig (Star Trek's Chekov) was in a couple of the episodes. Bits of it can be seen on Youtube. Episode 1 is called "Voyage of discovery."