The pair-correlation function g₂(r) describes short-range order in many-particle systems. It must obey two necessary conditions: (i) non-negativity for all distances r, and (ii) non-negativity of its associated structure factor S(k) for all k. For the elementary unit step-function g₂ form, previous work [F. H. Stillinger, S. Torquato, J. M. Eroles, and T. M. Truskett, J. Phys. Chem. B 105, 6592 (2001)] indicates that (i) and (ii) could be formally satisfied, but only up to a terminal density at which the covering fraction of particle exclusion diameters equaled 2^–d in d dimensions. To test whether the unit step g₂ is actually achievable in many-particle systems up to the apparent terminal density, a stochastic optimization procedure has been used to shift particles in large test systems toward this target g₂. Numerical calculations for d = 1 and 2 confirm that the step function g₂ is indeed realizable up to the terminal density, but with substantial deviation from the configurational preferences of equilibrium hard-rod and hard-disk models. We show that lineal statistical measures are particularly sensitive to this difference. Our results also illustrate the characteristics of "closest approach" to the step function g₂ above the terminal density.