Contractile tails are key components of the biological nanomachinery involved in cell membrane puncturing, where they provide a means to deliver molecules and ions inside cells. Two intriguing examples of contractile tails are those from bacteriophage T4 and R2-pyocin. Although the two systems are different in terms of biological activity, they share a fascinatingly similar injection mechanism, during which the tail sheaths of both systems contract from a so-called extended state to around half of their length (the contracted state), accompanied by release of elastic energy originally stored in the sheath. Despite the great prevalence and biomedical importance of contractile delivery systems, many fundamental details of their injection machinery and dynamics are still unknown. In this work, we calculate the bending and torsional stiffness constants of a helical tail sheath strand of bacteriophage T4 and R2-pyocin, in both extended and contracted states, using molecular dynamics simulations of about one-sixth of the entire sheath. Differences in stiffness constants between the two systems are rationalized by comparing their all-atom monomer structures, changes in sheath architecture on contraction, and differences in interstrand interactions. The calculated coefficients indicate that the T4 strand is stiffer for both bending and torsion than the corresponding R2-pyocin strands in both extended and contracted conformations. The sheath strands also have greater stiffness in the contracted state for both systems. As the main application of this study, we describe how the stiffness constants can be incorporated in a model to simulate the dynamics of contractile nanoinjection machineries.