In addition to what ballengerb1 pointed out (that it's gravity, not centripetal force), I should point out to you that astronauts and space stations are very much affected by gravity. As you rightly surmised, since they're only a few hundred miles from the earth's surface, gravity is almost as strong up there as it is a sea level. So why do they feel weightless? Because they are constantly falling toward the earth. They feel weightless for the same reason that a skydiver feels weightless when he or she first jumps out of an airplane (before wind resistance limits their acceleration). Because they're free-falling!
So if it's constantly falling, why doesn't the space station plummet to the ground? Because it's moving sideways so fast! Imagine for a minute that you're sitting in an orbiting space station looking directly down at the surface of the earth through a window, and imagine that gravity suddenly disappeared. The space station would continue onward in a straight line (parallel to the surface of the earth at the instant when gravity was shut off). At first it would seem like you were maintaining your altitude, but since the earth's surface is curved, not straight, you'd find that after a little while the ground beneath you would be getting further and further away. You'd be traveling in a straight line, but the curved earth below you would be "dropping away" until you reached the planet's edge, at which point you'd be looking "down" directly along the horizon a quarter of the way around the planet from where you were first released from gravity and some 7000km below you.
So now if you imagine back to the realistic scenario where there IS gravity, you can see that while the space station is falling toward the earth's surface, the earth's surface is simultaneously dropping out from underneath it as it moves sideways. The two counteract each other exactly, and the space station orbits around the earth at a nearly constant altitude. That's why the space station never actually falls to the ground, but yet its occupants feel weightless.
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