Just because something’s invisible doesn’t mean it’s weightless. So, no, air is not weightless.
As a matter of fact, all gases have mass, and therefore they have weight. Air, a combination of gases, is fairly heavy.
Good thing, too, the weight of the air around us and the force it exerts on our bodies pushes on us, and that is what keeps us from exploding. The pressure inside our bodies, in turn, keeps the air pressure from squishing us flat.
Air weighs about .07 pounds per cubic foot.
And believe it or not, each square inch of our bodies is under about 14.7 pounds of air pressure.