Why do nucleons feel a repulsive force when less than 1 fm? My Modern Physics textbook by Taylor states that when nucleons are less than 1 fm apart, there is a strong repulsive force between them. I am fairly certain that it is not the Pauli Exclusion Principle that the book is talking about. So if this is true, what is the mechanism for such a force?
 A: The repulsive core of the nuclear potential may be modeled by the ω(782 MeV) isosinglet vector meson exchange. See, e.g.,
Scholarpedia. 
Vector exchange, like the photon exchange between two electrons, is repulsive.
The long-range attractive part is due to the pion pseudoscalar exchange, π(138ΜeV).
Recall that in natural units, 1= 197 MeV fm.
Now, 0.8 fm is also the charge radius of the proton,  thus near the color confinement radius, where QCD internal nucleon interactions must be taken into consideration.
Nevertheless, damning/oversimplifying complications, the answer to you question is the ω effects the bulk of the nuclear potential core repulsion.
A: Nucleons get close to each other to form nuclei so this blanket statement of repulsion needs qualification and I am puzzled why the Pauli exclusion principle is involved.  
Protons will repulse protons at one fermi due to the same charge repulsion. That is the reason that in order to bind into nuclei of more than one nucleon neutrons are necessary so that the repulsive electric force is overcome by the much stronger attractive force of the strong interactions and a stable solution exists , which are black lines in this wikipedia diagram of nuclides. All the other isotopes decay because there exist lower energy states than the one bound by the strong interactions between the  nucleons (protons and neutrons). In general for high Z more than twice the number of neutrons are needed to overcome the same charge repulsion from the protons and get a stable nucleus.
The only place I have seen the Pauli exclusion principle invoked in connection to nucleons is in justifying why all the nucleons of a nucleus do not end up on one energy state, the lower one ( similar to why all the electrons in an atom do not end up in the lowest ground state), but fill up  consecutive energy layers.

The Pauli exclusion principle is involved in the basic explanation of the shell model for nuclear energy states. The evidence for shell structure in the nucleus was surprising at the outset, because a dense collection of strongly interacting particles should be bumping into each other all the time, resulting in redirection and perhaps loss of energy for the particles. The Pauli principle effectively blocks the loss of energy because only one nuclear particle can occupy a given energy state (with spin 1/2, neutrons and protons are fermions.) In this dense collection of matter, all the low energy states will fill up. This means that the particles cannot take part in interactions which would lower their energy, because there are no lower energy states they can go to. Scattering from an external particle which raises the energy of a nucleon can happen, but scattering which lowers an energy level is blocked by the exclusion principle. 

Now if he is talking of scatterings at high energy, proton proton scattering for example, that probe small distances, deep inelastic scattering showed the hard core existence of quarks. Again I have difficulty to think how the Pauli exclusion would contribute to this hard core since the energy states  in scattering are continuous and fermions could easily be accommodated in different energy states  but within the Heisenberg uncertainty principle indistinguishable.
A: Protons and neutrons are different
Protons contain one elementary charge and neutrons electric charges with no net charge. The consequence is that there exists an electric attraction between a proton and a neutron equilibrated by the magnetic repulsion between nucleons. The electromagnetic interactions between nucleons are, except for the Coulomb repulsion between protons completely ignored by the nuclear scientists.
