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.