Earth's gravitational field can indeed be felt close to the planet. The reason people are seen floating around in space stations is that they are orbiting around the earth. When set in orbit, the object's velocity around the Earth is set such that the gravitational pull of the planet (accelerating the object towards Earth) and the velocity of the orbiting object (in a direction that would actually make it fly past the planet) compensate each other's out, leading to a largely static orbit and a state of weightlessness, but certainly not a gravity-free environment.

Another question, though, is why is that done this way?]