We are just a bunch of molecules that hang out together but don't touch each other. Is this believable? If this is the case, what goes on in the body that allows us to "feel" when we touch something? How is this related to an object applying a force to another object?

While the idea that we are a bunch of molecules that do not ever "touch" may seem counterintuitive, it is indeed the case. If we think about the structure of an atom, it is made of a nucleus composed of protons and neutrons, and surrounded by electrons. The electrons are moving at very high speeds in random directions all around the nucleus creating electromagnetic fields, defined by various wave functions. These electron clouds are essentially what do the "touching" between molecules.

When two molecules come in contact with each other, rather than solid particles physically touching, it is the two electron clouds of each molecule that are interacting and essentially repelling each other. If we try to bring two negatively charged species together, they will naturally repel one another. Therefore, when two electron clouds try to interact, the electrons will electrostatically repel each other. When this occurs, the force that each electron cloud exerts electrostatically on the other cloud is transmitted to their nuclei. This is how it is related to general objects applying forces to one another. The force of repulsion felt by the electron clouds is ultimately transmitted to the nerve cells in our skin which is perceived as pressure and a sense of touch, despite the fact that no particles are physically touching.