Clouds

Perpetual cloud cover -- such as what you might find on Venus -- would of course obscure the night sky and the stars therein. The caveat here is that you would likely have people wondering "What's beyond the clouds?", and if any of those happen to get through them during the night they'd see the stars and then -- potentially, anyway -- cue the space race.

Light

It's well known that we can't see the stars (save for our own sun, which is of course a star itself) during the day. This is because the sun is so bright that it simply obscures the stars from view. At night, however, our own planet blocks the sun's light and allows us a view of the stars above.

We can see the stars at night because our moon doesn't reflect back enough of the sun's light to obscure the stars. This is because its albedo is only ~12%. However, if our moon were "brighter" -- say, closer to Saturn's moon Enceladus's 99% -- it could conceivably reflect enough light back toward the planet to make the night sky almost as bright as the day's, enough so to obscure the stars from view.

This would require a planetary system where the moon doesn't go through phases -- i.e. it orbits the planet at precisely the right period to keep it on the opposite side from the sun, but is never eclipsed (I don't even know if such a configuration is possible!) -- or where there are multiple high-albedo moons orbiting in such a way that there is always at least one moon bright enough to obscure the stars from view.

Alternatively, a binary (or more) solar system might be configured in a way that there's always at least one sun filling this role of obscuring the "night" sky and its view of the stars.

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