We study the mechanical and conformational properties of networks of helical polymers with a combination of Monte Carlo simulations based on the Wang-Landau algorithm and the Three-chain Model. We find that the stress-strain behavior of these networks has novel features not observed in typical networks made of synthetic polymers. In particular, we find that as these networks are stretched they first strengthen, then soften and finally strengthen again. This non-monotonic behavior of the stress correlates with the one of the helical content and is rationalized by the elongation-induced formation and melting of the helical structure of the polymer. We complement these results with a study of other conformational properties of the polymer strands that clarify the molecular mechanisms behind the mechanical behavior of these networks. Finally, we present a qualitative comparison of some of our results with the theoretical ones recently reported by Kutter and Terentjev.