I would tend to say that what you "feel" is not "force" per se but the
distortions they cause to your body.
* Gravity pulls on all parts of your body and hence doesn't cause any
distortion and hence you don't "feel' it. But if you are standing on
the floor, the soles of your feet compress, causing the sensation of
force.
* Air presses uniformly on your skin from all sides, which creates a
"typical" compression that your body ignores. If the pressure to a
particular spot either increases or decreases, that creates is a
distortion that registers as a force on that section of your body.
* If you hold an object in your hand, the area under that object gets
compressed and you "feel" the weight of the object.
* if a force remains constant long enough, the body becomes accustomed
to the new equilibrium, and starts to ignore the force. When you put a
hat on, you notice the forces, but after a while you can hardly notice
the hat. (that might partly explain why low density objects tend to
"feel" lighter than high density objects of the same mass - at least
they do to me. When the force is quite spread out, there is little
disortion to the skin and you quickly ignore the distortion. A more
concentrated distortion is harder to ignore.)