It was a serialized cartoon made up of short segments. The introduction to each segment featured a sort of clock superimposed on a sci-fi scene (a space station?), with a hand that counted down. There was a space ship. The drawing was fairly stylized, and suggestive of a remote, deep space setting. Movement was not sophisticated, and may have been a bit jerky, a la Clutch Cargo. I recall a vibe of mystery. It may have been broadcast as a part of another show, perhaps a childrens show. Captain Kangaroo comes to mind, but I don’t think that was it for some reason. I would guess it was 1960s-era, but not sure.
The mention of "Clutch Cargo" suggests "Space Angel":
Space Angel is an animated science fiction television series produced in the United States from early 1962 through 1964. It used the same Synchro-Vox lip technique as Clutch Cargo, the first cartoon produced by the same studio, Cambria Productions.
The series chronicled the adventures of three astronauts who worked for the Earth Bureau of Investigation's Interplanetary Space Force on board the spaceship Starduster: Captain/Pilot Scott McCloud, also known as "The Space Angel" (voiced by Ned Lefebver), Electronics/Communications expert Crystal Mace (voiced by Margaret Kerry), and the immensely strong Scottish born Gunner/Engineer Taurus (voiced by Hal Smith).
I have a very remote memory of this cartoon, Johnny Cypher in Dimension Zero, which was triggered by the image of Space Angel linked by the other answer (image from The Big Cartoon Database, while here you can see the images of the other characters) .
Unluckily I remember just the far-space setting and the 60s style, but according to IMDb:
The human scientist Dr. Johnny Cypher (Paul Hecht) is possessed of super-humanoid powers (the origins of which are never disclosed) that allow him to travel between continuums of time, outer space, and inner space through "Dimension Zero"
Which could explain the clock you remember at the beginning of the episodes.
In the year 2199, an alien race known as the Gamilas (Gamilons in the English Star Blazers dub) unleash radioactive meteorite bombs on Earth, rendering the planet's surface uninhabitable. Humanity has retreated into deep underground cities, but the radioactivity is slowly affecting them as well, with humanity's extinction estimated in one year.
I remember that at the beginning of every episode, there was a kind of countdown telling how much time was left to mankind. Of course, it wasn't composed of short episodes, so maybe it is not what you are looking for.