In this paper, a novel concept called simultaneously transmitting and reflecting RIS (STAR-RIS) is introduced into the wireless-powered mobile edge computing (MEC) systems to improve the efficiency of energy transfer and task offloading. Compared with traditional reflecting-only RIS, STAR-RIS extends the half-space coverage to full-space coverage by simultaneously transmitting and reflecting incident signals, and also provides new degrees-of-freedom (DoFs) for manipulating signal propagation. We aim to maximize the total computation rate of all users, where the energy transfer time, transmit power and CPU frequencies of users, and the configuration design of STAR-RIS are jointly optimized. Considering the characteristics of STAR-RIS, three operating protocols, namely energy splitting (ES), mode switching (MS), and time splitting (TS) are studied, respectively. For the ES protocol, based on the penalty method, successive convex approximation (SCA), and the linear search method, an iterative algorithm is proposed to solve the formulated non-convex problem. Then, the proposed algorithm for ES protocol is extended to solve the MS and TS problems. Simulation results illustrate that the STAR-RIS outperforms traditional reflecting/transmitting-only RIS. More importantly, the TS protocol can achieve the largest computation rate among the three operating protocols of STAR-RIS.