Yes, air does exert pressure on us.
The air particles around you exert a certain amount of pressure on your entire body, but what’s interesting is that the same amount of pressure is exerted back onto the air molecules by the insides of the body, thereby achieving a state of equilibrium.
The pressure of air that is present outside your body is the same as that of the air present ‘inside’ your body. The air that is constantly present in your lungs, ears and nose has the same atmospheric pressure as the air on the outside of your ears, nose, and chest. Since there is no pressure difference, we don’t feel anything at all, as far as atmospheric pressure is concerned.