No, our weight is what keeps us on the ground.  Without gravity, mass would not know which way is down.

Weight does keep us on the ground, but there's a distinction between weight and mass. An object like a large meteoroid has a "mass" but would have a different weight on Earth than on the Moon, as the Earth has greater attracting strength, so its weight will very depending on the proximity to another mass.
The Earth (whether globe or pancake) has mass but no weight as it is not resting on another mass. If it could rest on the surface of the Sun it would have weight.
An object free-falling off a cliff has no weight, but of cause mass.