Gravity does exist in space. However, objects inside of a spaceship are experiencing the same force of gravity as the ship itself does, so they feel no relative gravity between themselves and the ship. The ship, however, still moves in a gravity-bound fashion. The reason why we actually feel gravity while standing on the Earth is that the Earth is pulling on us and we are actually standing on it, not falling toward it or orbiting it. We don't, for example, feel the gravity of the Sun while standing on the Earth - that is because we, like the Earth, are orbiting the Sun together. The same goes for a spaceship orbiting the Earth.
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