Why Pauli exclusion instead of electrons canceling out? To quote Wikipedia,

The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions (particles with half-integer spin) may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles.

As far as I can tell, this means that two identical electrons in identical states have wave functions that cancel each other out.
My question is, if this is the case, why do we have the Pauli exclusion principle? It seems that getting two electrons too close should annihilate both of them (or cancel them out, or however you want to say it), but instead we have the Pauli exclusion principle, where somehow the electrons remain separate. Why?
 A: It seems that getting two electrons too close should annihilate both of them.

Your assumption is wrong. Only antimatter cancels out matter. In case of Electron, only a Positron can cancel it out. Mind it, two electrons repel each other due to identical charges (in classical sense).
In true sense, you shouldn't talk about distance. To share same quantum state, electrons needs to be different by any attribute. That's why electrons having different Spins can share same quantum state.
A: Putting two fermions in the same state doesn't cause the two fermions to cancel out and disappear; it causes the entire universe to cancel out and disappear. Or, more prosaically but accurately, a state with two fermions in the same single-particle state simply isn't a valid state in the fermionic Hilbert space. The point is that the badness isn't just isolated to the two fermions themselves - it incurably corrupts the entire system. Fermions are weird, only semi-local beasts.
However, there are particles that behave in exactly the way you're describing: putting two of them in the same state causes them to cancel each other out, but leaves the rest of the system completely unharmed. They're misleading called "Majorana fermions" in the condensed-matter community, although this name is extremely unfortunate, because (a) they're not actually fermions, but rather "anyons", and (b) they're not the same kind of particle that high-energy physicists refer to as "Majorana fermions", which really are fermions.
A: Feynman in multiple writings suggested thinking about "exchanging particles" in terms of exchanging them as they move through time. That is, they can either move in two parallel paths as they move forward, or they can cross paths (exchange roles).
The antisymmetric cancellation applies to the latter, but not to the former. Now if you think that through, it means that the parallel path remains strong even as the crossover paths cancel out, resulting in the two particles avoiding each other and maintaining unique paths (wave functions). The net result is not full cancellation, but cancellation at the edges, where the particles would cross. (Feynman goes into a lot more detail about rotations, but frankly that part can get you sidetracked a bit; it's the "anti-crossover" part that counts in terms of actual outcomes.)
Another consequence of identical fermions cancelling each other out is that packing more fermions into a tight space forces their space-filling wavelengths to become shorter also. Since in quantum mechanics the spatial wavelength of a particle defines its momentum, particles that are squeezed in this fashion also get very, very hot.
A neutron star is a good example. Pauli exclusion -- the "constriction of space because crossover cancels but parallel does not" -- allows neutrons to pack together very densely indeed.
There are limits, however. When gravity gets too monumental, even Pauli exclusion is unable to keep up with the pace, and the entire star collapses, very quickly. Thus is born a stellar-sized black hole, or at least this is one example of how one can form.
