I've heard this question asked before and I usually take it as "how do you stop gravity from acting on something?" They sometimes compare it to wrapping something in aluminum foil to shield it from EM radiation (which is generally effective.) I think the idea comes from the thought that a gravity "field" must be a lot like an electromagnetic field since they are both fields.
When a lot of people think of shielding gravity, I believe that they are thinking along the lines of something like Wells' "Cavorite" from
First Men in the Moon. I seriously doubt that this type of shield is possible.
Something along the lines of passive magnetic shielding is also brought up. However, this type of shielding works by redirecting the lines of force around the shielded area rather than blocking it. The lines are concentrated into the shielding material. Even if possible, this seems to me would lead to the shield's weight to increase to compensate for the decrease in weight for whatever is being shielded. IOW, you wouldn't be able to encase an object in this shield and have the combination be weightless.
The other possibility is to have your shield in place, and place an object over it. However, I would think that this would be an unstable position as the shield would only be effective on that gravity coming from directly below and unless the object stayed in the exact center, it would drift towards the edge like a ball rolling off the peak of a hill.
Then there's the conservation of energy issue. In moving your object from outside the shielded area to the center, you are in effect moving it from one gravitational potential to another. IOW, it should take as much energy to do so as it would to lift that object to an infinite distance from the Earth or accelerate it up to escape velocity. In addition, as that object drifts away from the center it will gain that energy back as kinetic energy accelerating as it does so, and when it hits the edge of the shielded region would be moving at escape velocity.
Any type of active shielding would have the same energy conservation issue; to render any object weightless, it would have to consume at least as much energy as it would to take to get that object up to escape velocity.