The ability of animal cells to crawl, change their shape, and respond to applied force is due to their cytoskeleton: A dynamic, cross-linked network of actin protein filaments and myosin motors. How these building blocks assemble to give rise to cells’ mechanics and behavior remains poorly understood. Using active micropost array detectors containing magnetic actuators, we have characterized the mechanics and fluctuations of cells’ actomyosin cortex and stress fiber network in detail. Here, we find that both structures display remarkably consistent power law viscoelastic behavior along with highly intermittent fluctuations with fat-tailed distributions of amplitudes. Notably, this motion in the cortex is dominated by occasional large, step-like displacement events, with a spatial extent of several micrometers. Overall, our findings for the cortex appear contrary to the predictions of a recent active gel model, while suggesting that different actomyosin contractile units act in a highly collective and cooperative manner. We hypothesize that cells’ actomyosin components robustly self-organize into marginally stable, plastic networks that give cells’ their unique biomechanical properties.