The parameters of thin-walled mechanical energy storage devices (flywheels) reinforced with high-strength threads using the continuous winding method are being calculated. Reinforcement is performed along so-called pseudo-geodesic trajectories, in which there are no interlayer stresses. The mathematical model of a composite material is accepted as a heterogeneous structure consisting of high-strength continuous reinforcing threads (tapes) of high rigidity and a relatively low-strength matrix (binder). The ratios of moment-free shells and the flexible thread theory, built on the basis of network analysis, are used. The shapes of the meridians of the shells, the angles of the reinforcement trajectories, and the magnitude of the axial thrust forces are determined. The technological possibility of reinforcing the shell using the continuous winding method is being investigated, in which the tape does not slip off the mandrel during the winding process. The specific mass energy intensity of flywheels is calculated. The obtained energy intensity values are compared with the energy intensity of equally stressed flywheels with equal parameters, and it is found that flywheels with pseudo-geodesic reinforcement are less energy-intensive. However, since the limitation on the strength of the binder does not apply to them, they have a wider range of existence of meridian shapes than flywheels of equal strength.