Many active fluid systems encountered in biology are set in total geometric confinement. Cytoplasmic streaming in plant cells is a prominent and ubiquitous example, in which cargo-carrying molecular motors move along polymer filaments and generate coherent cell-scale flow. When filaments are not fixed to the cell periphery, a situation found both in vivo and in vitro, we observe that the basic dynamics of streaming are closely related to those of a nonmotile stresslet suspension. Under this model, it is demonstrated that confinement makes possible a stable circulating state; a linear stability analysis reveals an activity threshold for spontaneous autocirculation. Numerical analysis of the longtime behavior reveals a phenomenon akin to defect separation in nematic liquid crystals and a high-activity bifurcation to an oscillatory regime.