Oh, but we do! In fact, we could say that there isn't any other kind of force that we can feel.
For forces that are visibly electrical and very feelable, have a look through the pictures of the demonstrations on this site. One of my favorites is the Van de Graaff generator which is used to make peoples' hair stand on end. Each hair, charged up like its neighbors, repels its neighbors due to electrostatic forces. I often feel my hairs stand on end if I get up out of a cheap plastic chair on a dry day. You can rub a balloon in your hair and get a similar effect. Discharging static electricity by touching a metal object (preferably one connected to the ground) makes a sometimes painful spark. I can feel that!
The forces between atoms and molecules in a substance are electromagnetic in nature. Pushing and pulling with contact forces really amounts to an electromagnetic interaction at the microscopic level.
I guess there's gravity, but if that's the only force on you, you won't feel it (you'd be in free-fall). If you were out in space and the only force on you is a push from some object in contact with you, you'd feel that one, so somehow gravity is odd-man out on that one. We could argue that you never feel gravitational forces, you only feel the forces supplied by the floor, the ground, or chairs needed to hold you up and keep you from falling.
(published on 10/22/2007)