Our understanding of how large-scale structure and dynamics emerge away from equilibrium is incomplete. We describe a hierarchical self-organizing process of remarkable complexity. Starting with an initially uniform mixture of microtubule-based active fluid and passive actin filaments, we observe the emergence of complex network structures and dynamical patterns on length scales ranging from nanometers to millimeters. Active fluids sculpt the structure, shape, mechanics, and dynamics of the actin network. Eventually, one observes the formation of macroscopically large actin-based thermalized solid-like membranes whose out-of-plane bending rigidity and in-plane oscillatory patterns are driven by the enveloping actin fluid. Taken together, these experiments demonstrate a need for developing a theoretical understanding of out-of-equilibrium self-organizing processes of soft networks.