The experimentally observed asymptotic scaling phi(L/D) constant, for the random packing volume fraction (phi) of rods with high aspect ratio (L/D), is shown to be the consequence of uncorrelated rod-rod contacts, the constant being equal to [c] = 5.4 +/- 0.2 contacts per rod. The weak dependence of [c] on the particle shape accounts for the drastic decrease in random packing density going from spheres to thin rods. Moreover, [c] approximate to 5.4 is large enough for isotropic rod packings to be metastable "rod-glasses" with respect to a fully ordered nematic phase. In addition to rod packings, literature results on colloidal sediments, rheology, and percolation of random rods are (re)interpreted to conclude that uncorrelated pair contacts suffice to explain densities of a variety of(mechanical) thin-rod systems.