Questions surrounding the spatial disposition of particles in various condensed-matter systems continue to pose many theoretical challenges. This paper explores the geometric availability of amorphous many-particle configurations that conform to a given pair correlation function g(r). Such a study is required to observe the basic constraints of non-negativity for g(r) as well as for its structure factor S(k). The hard sphere case receives special attention, to help identify what qualitative features play significant roles in determining upper limits to maximum amorphous packing densities. For that purpose, a five-parameter test family of g's has been considered, which incorporates the known features of core exclusion, contact pairs, and damped oscillatory short-range order beyond contact. Numerical optimization over this five-parameter set produces a maximum-packing value for the fraction of covered volume, and about 5.8 for the mean contact number, both of which are within the range of previous experimental and simulational packing results. However, the corresponding maximum-density g(r) and S(k) display some unexpected characteristics. These include absence of any pairs at about 1.4 times the sphere collision diameter, and a surprisingly large magnitude for S(k=0), the measure of macroscopic-distance-scale density variations. On the basis of these results, we conclude that restoration of more subtle features to the test-function family of g's (i.e., a split second peak, and a jump discontinuity at twice the collision diameter) will remove these unusual characteristics, while presumably increasing the maximum density slightly. A byproduct of our investigation is a lower bound on the maximum density for random sphere packings in d dimensions, which is sharper than a well-known lower bound for regular lattice packings for d greater than or equal to 3.