Would it be possible for a lattice of protons and neutrons (I'm picturing a plane of hexagons in my head) to exist bound by the strong nuclear force (not gravity)? I know that the strong force losses its power when an atomic nucleus gets to be too large, but in a lattice, it would only have to bond one proton to a few neutrons or one neutron to a few protons at a time. Would this work? If so, what would be the properties of such a material?
No, because the repulsive power of the protons accumulates toward infinity, blowing the structure apart. To accomplish what you have described you would need stable, negatively charged particles like antiprotons... which of course also fail in such an arrangement because they annihilate the protons!
The instability from accumulating positive charge is also why higher elements become unstable.
You can create crystalline structures that are composed mostly of neutrons with electron-neutralized protons sprinkled in, but only if you have very powerful compression to keep the neutrons stable and the whole arrangement bound together. That's the material in neutrons stars, sometimes called "neutronium" (yes, as in Star Trek), and it's very, very stiff. Breaks in the neutron crusts of such stars lead to abrupt changes in the otherwise incredibly precise timing of pulsar radio bursts.
A lone proton in between a crystal lattice could find itself with more negative charge then equilibrium from the neighbouring clouds of electrons. Thus there is a compressive force driven by the lattice and heat, in addition to proton repulsion of the nuclear force.
I expect over time their will be a determination that the coulomb barrier may not be static under all conditions and because it is not yet understood it's difficult to replicate working Low Energy Nuclear reactions. However the anomalous results show something is happening. That can only mean that the understanding is incomplete based on observations to date.