Despite their long history in experiment, simulation, and analytical theory, pair correlation functions that describe local order in many-body systems still retain a legacy of mathematical mysteries. One such open question concerns "realizability" of a given candidate pair correlation function, namely whether it actually represents the pair correlation for some spatial distribution of particles at number density rho >0. Several necessary conditions that must be satisfied by the candidate are known, including nonnegativity of the function and its associated structure factor, as well as constraints on implied local density fluctuations. However, general conditions sufficient to ensure realizability are not known. To clarify this situation, we have examined realizability for a simple one-dimensional lattice model, with single-site occupancy, and nearest-neighbor exclusion. By virtue of exhaustive enumeration for systems of 15 or fewer sites subject to periodic boundary conditions, several conclusions have been formulated for the case of a constant pair correlation beyond the exclusion range. These include (a) pair correlation realizability over a nonzero density range, (b) violation of the Kirkwood superposition approximation for many such realizations, and (c) inappropriateness of the so-called "reverse Monte Carlo" method that uses a candidate pair correlation function as a means to suggest typical many-body configurations.