While contraction of muscle actomyosin assemblies is well understood, this is not the case for disordered networks of actin filaments (F-actin) driving diverse essential processes in animal cells. For example, at the onset of meiosis in starfish oocytes a contractile actin network forms in the nuclear region transporting embedded chromosomes to the assembling microtubule spindle. Interestingly, we find no evidence of myosin activity driving network contractility. Here, we addressed the mechanism driving contraction of this 3D disordered F-actin network by comparing quantitative observations to a continuous model and simulations of the actin network. In both formulations, only a disassembly-driven mechanism correctly predicts perturbation experiments in which actin is stabilized or destabilized. We propose a mechanism in which the depolymerization energy is harnessed to provide contractile work.