The Hooke-Calogero model of the H2 molecule, a four body system consisting of two protons and two electrons with harmonic electron-nuclear, Coulombic electron-electron and inverse quadratic nuclear-nuclear interactions, has been analyzed in-depth. A fully analytical closed-form non-Born-Oppenheimer solution has been found and based on it the properties of both electron-pair and nuclear-pair densities have been studied. Nuclei have been found to be strongly correlated in a way that resembles a sort of Wigner crystallization, for moderate electron-nucleus confinement strengths. Additionally, we have explicitly evaluated the Coulomb holes for the electrons. Analysis of these holes reveals that the similarity between the electron correlation effects of the model as compared to the real Coulombic systems is remarkable.