Zero gravity - how it is possible? If planets in the solar system are under the influence of gravity, how does one achieve zero gravity just outside Earth? (Please correct the question if the terms are wrong, but I hope the question is clear.)
 A: The planets outside the earth's atmosphere are too far away and too low in mass to create a substantial effect on say, the International Space Station. The Earth is still pulling down on the ISS, but the ISS is also falling in a circle around the Earth.
So the zero gravity effect is not caused by removing gravity. If you were silly enough to jump off the roof of your house, you would feel weightless until you hit the ground. 


In these two photos you can see that the people are floating because the plane is diving/ falling, they are not resisting gravity. The zero gravity effect in space is pretty much the same idea. 
It is only when you have to withstand the force of gravity, by standing up, that you feel it's effect. Gravity does not and will not go away,  but if you can temporarily resist "fighting" it's effects, it feels like you are weightless.
As the ISS falls around the Earth, it is not resisting gravity, and therefore people inside do not feel a gravitational force.
A: You said, "outside earth". That's a debatable term. Anything raised above the earth's that's not touching it, is technically outside earth, unless you take atmosphere into account. Anyway, I think you mean something thousands of miles away from earth's surface. Moon? It's still has gravitational interactions with earth. I think you mean an astronaut in orbit feeling weightlessness, or an object far away from earth. Gravity works on the principle or inverse-square law when it comes to distance(note we are talking about Newtonian concept of gravity). Force of attraction is also dependent on how massive the other body is, so the force of gravity on Moon due to earth won't be same as that on a tennis ball at the same distance.
Also, greater the distance, lesser with be the force of attraction. So, a body(lets say a tennis ball) raised very high above the surface of the earth's surface(millions of miles), won't be experiencing the same force as the one on earth, that means the value of g on it has reduced due to the huge distance, but it's still under the influence of earth's gravity, unless the distance is so vast that force becomes theoretically negligible. The body is still attracted towards earth, but not with an acceleration of g. 
If it gets closer to another planet or asteroid, the influence of that body on the tennis ball would be higher.
A: No one has achieved zero gravity. Astronauts experience zero gravity because the force of gravity is balanced by centrifugal force of orbital velocity. The gravity is not really zero in the orbit, only it is balanced by the motion.
Now somewhere between earth and moon, one can find a region where gravity of both balance each other's. So, that place will appear to be a zero gravity. But it is not, because anything placed in that region will continue to rotate around sun along with earth and moon due to sun's gravity, and so not zero gravity there as well.
Same way, if you find such a region between earth and sun, they both keep rotating around center of our galaxy. And that regoin is no zero gravity.
It will be very hard to find a place where gravity is zero, but if you go sufficiently far from the nearest galaxy, that place may be pretty close to but not exactly zero gravity.
A: There is no such thing as zero gravity. How could you ever find a place in the universe where there was no gravity? The galaxies and stars would always have their effect. Granted it may be a feeble force in places, but it would be there.
