The connectedness percolation threshold (eta(c)) and critical coordination number (Z(c)) of systems of penetrable spherocylinders characterized by a length polydispersity are studied by way of Monte Carlo simulations for several aspect ratio distributions. We find that (i) eta(c) is a nearly universal function of the weight-averaged aspect ratio, with an approximate inverse dependence that extends to aspect ratios that are well below the slender rod limit and (ii) that percolation of impenetrable spherocylinders displays a similar quasiuniversal behavior. For systems with a sufficiently high degree of polydispersity, we find that Z(c) can become smaller than unity, in analogy with observations reported for generalized and complex networks. DOI: 10.1103/PhysRevLett.110.015701