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.
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 (though as Sean Raymond points out in comments this is probably impossible, as this would pretty much have to put it in the L2 Lagrange point, which is unstable) -- or where there are multiple high-albedo moons orbiting in such a way that there is always at least one moon (or combination of two or more) 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.
But wouldn't a moon give the inhabitants reason to want to head into space? It might, but on the other hand remember that we're talking about moons that are nearly as bright as the sun itself, which makes direct observation very very difficult (unless you like your retinas burned off). Without direct observations, determining that the moons are something that could be visited would be fraught with unique difficulties, to say nothing of the fact that there's no observable universe for the inhabitants to see and from that to derive relative scales, such as the sizes and distances of their moons and sun; they may never even arrive at concepts such as "orbits", something we were able to figure out after millennia of studying the other planets in our solar system.
Now, granted, you might be able to make some observations of a crescent moon, but the same high albedo that obscure the stars themselves would likewise obscure the "dark side" of the moons. Still, the fact that the moons go through phases while the sun doesn't will naturally drive curiosity and could indeed lead to someone figuring out that they're orbiting bodies they might be able to reach, but again without any concept of relative scale derived from observations other bodies, to say nothing of the complete absence of even a rudimentary grasp of orbital mechanics, the challenges in actually getting out there may prove insurmountable, or at the very least dissuade anyone who might have been capable of surmounting them.