We consider a network of interconnected dynamical subsystems with a state-space safety constraint and propose a verification technique that constructs a (robustly) invariant set verifying safety. The invariant set is a sublevel set of a Lyapunov function constructed from local storage functions for each subsystem. Our approach requires only knowledge of a local passivity property for each subsystem and the static interconnection matrix for the network, and we pose the safety verification as a sum-of-squares (SOS) feasibility problem. We consider first the case when, in the absence of disturbance, the unique equilibrium of the network is known. We then extend these results to the case when the equilibrium of the networked system is unknown.