This is a touchy subject. There have been hundreds of well-funded projects into anti-gravity. But to date, there has been no proven progress - regardless of some wacky claims.

Gravity is an electromagnetic force. Gravity can and has been manipulated to a very small degree, but not completely overcome.

Lets say you have trillions of Nanobots designed to individually analyse the force of gravity and apply an opposing magnetic field of the same or greater strength. Since the Nanobots are so small, the force of gravity on the molecules that make up the Nanobots will be small too. This means it would require much less energy to oppose gravity using Nanobots than it would for a bigger machine.

For example, if you had trillions of Anti-Gravity Nanobots packed tightly and thinly onto a small area, anything placed above the Nanobots would be shielded from Gravity, and therefore float... in theory.