In space we feel weightlessness because the earth's gravity has less effect on us. Why do we not see the effect of the gravitational force between the various objects in a spacecraft? We see them floating around. Since the objects in a spacecraft are comparatively close to each other we should be able to see the gravitational effect between them.
Although the Earth's gravity has a lesser effect on an astronaut orbiting the Earth in a spaceship than on a person on the surface of the Earth, this is not the reason why an astronaut experiences weightlessness. The space shuttle, International Space Station and most other manned vehicles don't get that far from the Earth. The Earth's gravitational attraction at those altitudes is only about 11% less than it is at the Earth's surface. If you had a ladder that could reach as high as the shuttle's orbit, your weight would be 11% less at the top. Put another way, a person who weighs 100 pounds on the Earth's surface would weigh about 89 pounds at the top of the ladder.
The reason why the person wouldn't feel weightless is because they are being pushed by the ladder - it is keeping them from falling. If they were to jump off the ladder, then they would feel weightless, at least up until the time they splatted on the ground. This is why astronauts feel weightless. The astronaut, the spaceship and everything inside it are falling towards the Earth. The reason why the astronaut doesn't go splat is because the Earth is curved and the astronaut, the spaceship and everything inside it are moving 'sideways' fast enough that, as they fall towards the Earth, the surface of the Earth curves away from them. They are always falling towards the Earth, but they never get there.
The reason why you don't see gravitational effects between objects in a spacecraft is because gravity is a very, very weak force. Of the four basic forces that scientists are sure about, gravity is, by far, the weakest one. Have you ever tripped and fallen down? Well, it took the whole planet to do that to you. Have you ever seen a sock stick to a shirt after it has come out of a dryer? That static cling, created by a slight imbalance of charge between the sock and the shirt, is stronger than the gravitational attraction of the Earth. The gravitational attraction between two small objects in a spacecraft would be overwhelmed by other forces, such as the force of the air being circulated throughout the spacecraft. Although the force of attraction is there, it is so weak that special care would have to be taken to notice it.
Steve Gagnon, Science Education Specialist (Other answers by Steve Gagnon)