That depends on what you mean by “this stuff” (and also on what you mean by “fully understand”). Most decent theoretical physicists do understand special relativity (the domain of E = mc^2), at least as well as anyone understands anything. It’s really not nearly as difficult or complicated as most laymen think. At the same time, though, everyone, even the greatest of geniuses, has something they don’t understand. Fortunately, the things people don’t understand aren’t constant from person to person, so a group of people can understand more than any single person in the group… but even then, there are some things that nobody understands.
And back to that point about “…at least as well as anyone understands anything”: A large part of science is realizing that the things we thought we understood, we actually don’t. Feynman talked a lot about this: He pointed out that, when describing the forces between protons and electrons, a person might say “It’s like a magnet”, and think that by saying that, they have a greater understanding of the forces between protons and electrons. But really, a magnet is an emergent result of the forces between sextillions of protons and electrons, and is therefore far more complicated than the force between a single proton and a single electron. If you don’t understand the force between a single proton and a single electron, then you certainly don’t understand the magnet, and trying to explain something you understand poorly by analogy with something that you understand even more poorly is an exercise in futility.
Addendum: I found the Feynman interview I was thinking of. I slightly misremembered it-- The actual analogy he talked about wasn’t talking about forces between subatomic particles in terms of magnets, but talking about magnets in terms of rubber bands. The underlying message is the same, though.
The idea of matter being “condensed” or “coagulated” energy is probably a good conceptual notion, and the idea of c^2 being a damn big number gives us an idea of just how dense that energy content is.
Another conceptual notion is to consider that a kilogram of uranium contain approximately 2.52x10^24 atoms, which is a hell of a lot of atoms, and uranium is a very heavy element – a kilogram of most elements would contain far more atoms. So an atom exists on a scale so tiny that it’s unimaginable to us, and no ordinary measurements apply. Yet the force binding the nucleus together against the force trying to tear it apart, if we could directly measure it in a single atom, would be on the order of several pounds.
A related conceptual notion is to consider that a kilogram of uranium contains approximately 2.52x10^24 atoms, and uranium is a very heavy element – a kilogram of most elements would contain far more atoms. So an atom exists on a scale so tiny that it’s unimaginable to us, and ordinary measurements of its size and mass are conceptually meaningless to us. Yet the force binding the nucleus together against the force trying to tear it apart, if we could directly measure it in a single atom, would be on the order of several pounds.
Nobody’s addressed this, but it’s extremely important.
The answer to why graphene doesn’t pass through a hand is the same reason that the hand doesn’t pass through a chair. It’s not the matter, or the thickness or thinness of the matter. “Matter” is volumetrically a tiny percentage of stuff. If matter were all there were, then stuff would pass through stuff - and with as much ease as the earth passing through space without getting hit by anything of equal size.
The rest of “empty” space is full of fields. The electromagnetic field is produced by charged particles, and atoms, therefore stuff, are full of charged particles in constant movement. The field prevents stuff from interpenetrating, much more so than matter.
Scientists do understand this, very well. The study of fields - the gravitational field, the Higgs field, the quantum fields, and the rest - is probably more important than the study of matter. Fields exist at every point of time and space; matter does not.
For that matter, the matter itself is all fields, too. Electrons are excitations of the electron field, up quarks are excitations of the up field, and so on.
Fundamentally the only reason that a pen placed on a table doesn’t fall straight through it onto the floor (or maybe through the floor, too, into the basement) is entirely probabilistic. It happens routinely at the quantum scale. It’s only at some arbitrary macroscopic scale that we just consider it “impossible”.