Did I get hit by a car this morning? Quantum mechanics allows us to describe a particle as a wave, and also a collection of particles, which a car happens to be. What separates a typical wave from a classical particle is that the position is not well defined for a wave.
I think that it is hard to tell exactly how big an atom is, because the electron of the atom may be anywhere, but the probability of finding the electron gets very small as you look far away from the nuclei. I think in some sense the atom is as large as we want, depending on the definition.
When a car drives past me, is the position of the car separate from my position in a quantum mechanical way, or is it correct to say that "part" of the car hit "part" of me? I understand that no matter the answer the force from the hit is so small that it is negligible in every way, but that is not the core of the question.
In addition, if I made any wrong claims, please correct me, thanks in advance.
 A: Asher's comment written above is simply wrong, and the reason is rather fundamental in quantum mechanics. "The car slightly hits you" isn't how it works in quantum mechanics.
The reason is that weak effects – such as very small but nonzero values of the wave function of an electron very far from the nucleus – do not cause tiny but observable effects like they do in classical physics.
Instead, the wave function has a probabilistic interpretation. So its being nonzero very far from the nucleus means that there is a small probability that a finite effect takes place.
So in quantum mechanics, you can ask whether a car hit you – whether some reaction took place. In quantum mechanics, questions about physical systems (including this one) have to be answered by measurements, otherwise they are physically and scientifically meaningless.
And quantum mechanics predicts some probabilities for one outcome of the measurement or another. If only 0.0001% of the integrated $\int |\psi|^2$ of the electron was located at the distance $R$ equal to the distance of your body from the car, then it means that there was a 99.9999% probability that no interaction has taken place at all.
On the other hand, there is a 0.0001% probability that some interaction did take place, but it wouldn't be a small interaction. It would be a rather large one, such as the ionization of one atom – of the car or your body.
Again, in quantum mechanics, small wave functions don't mean small forces. They mean finite (not so small) jumps happening with tiny probabilities.
Again, it's very likely that you didn't get hit by a car in the morning at all.
One can give as many serious examples of that as we want. For example, when a radioactive nucleus decays, the wave function for an alpha particle gradually grows outside the nucleus. It doesn't mean that the alpha particle is "weakly felt" at all times. Instead, it means that there is some probability that the nucleus has decayed up to some moment, and some probability that it remained intact.
Also, one may consider collisions of particles, such as those at the LHC. Various quantum processes may take place; they are described by Feynman diagrams. The two colliding protons have an infinite cross section to interact through the electrostatic repulsion. So yes, there was always some interaction of this electrostatic kind. It's equivalently interpreted as the certainty that some very soft photons are being emitted. (One could say that the car-body contact above was electrostatic elastic between two electrons as well, in that case, it would always occur.)
But there are also other, short-range forces, that predict finite cross sections. In some reactions, some finite process like that – accompanied by the production of pions etc. – can take place. But in others, it strictly doesn't take place. The atoms effectively interact e.g. with van der Waals forces that decrease as a faster power law of the distance and they have a finite cross section. So when we talk about similar interactions between the whole atoms, then the interaction probably didn't occur at all – strictly zero at a probability that is almost 100%.
