We apply a recently developed density functional theory for freely hinged, hard polymeric chains to calculate the elastic moduli of an idealized polymeric solid lacking long-range bond order. We find that for such a model packing effects dominate the elastic behavior of the polymeric solid in a similar way as is the case in the hard-sphere crystal, which we reexamine. Our calculations show that the elastic stiffness of the model polymeric solid is essentially determined by how far one is removed from its melting point. The main role of the chain connectivity is to destabilize the solid relative to the equivalent solid of hard monomers. Comparison of our results with experimental data on semicrystalline polymers shows order-of-magnitude agreement.