Wireless powered and backscattering mobile edge computing (WPB-MEC) network is a novel network paradigm to supply energy supplies and computing resource to wireless sensors (WSs). However, its performance is seriously affected by severe attenuations and inappropriate assumptions of infinite computing capability at the hybrid access point (HAP). To address the above issues, in this paper, we propose a simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) aided scheme for boosting the performance of WPB-MEC network under the constraint of finite computing capability. Specifically, energy-constrained WSs are able to offload tasks actively or passively from them to the HAP. In this process, the STAR-RIS is utilized to improve the quantity of harvested energy and strengthen the offloading efficiency by adapting its operating protocols. We then maximize the sum computational bits (SCBs) under the finite computing capability constraint. To handle the solving challenges, we first present interesting results in closed-form and then design a block coordinate descent (BCD) based algorithm, ensuring a near-optimal solution. Finally, simulation results are provided to confirm that our proposed scheme can improve the SCBs by 9.9 times compared to the local computing only scheme.