We study the random close packing of a binary mixture of spheres and rod-like particles (spherocylinders) by the mechanical contraction computer simulation. We investigate the universality in packing of near-spheres by monitoring the position and the value of the maximum in the mixture packing density as a function of the mixture composition and the rod aspect ratio. We find that independently of the mixture composition the particles pack more efficiently/densely as the rod aspect ratio is perturbed slightly from zero and the maximum density is always reached at one unique rod aspect ratio of about 0.45. The dependence of the value of the maximum packing fraction on the mixture composition (the relative rod volume fraction) is linear, exhibiting some ideality in packing of near-spheres. This counter-intuitive finding suggests that even for high rod concentrations in a rod-sphere mixture the packing is governed by local contacts between the neighboring particles, which is usually observed for dilute colloidal suspensions and granular gases, where there is no correlation between the particles. The plausible explanation for this intriguing behavior is that the correlations between the particles are completely lost in the range of distances of several particle diameters, which can be originated from the decoupling of the orientational and translational degrees of freedom of the nearly spherical rods. This gives rise to the universality and locality of random close packing of the rod-sphere mixtures.