The source and destination should be located on the same side of reconfigurable intelligent surface (RIS), which limits its flexibility and coverage. In this correspondence, a simultaneously transmitting and reflecting RIS (STAR-RIS) aided full-duplex non-orthogonal multiple access scheme is proposed to structure a full-space smart radio environment. By jointly optimizing the power allocation coefficients at the access point, the transmit beamforming at the cell-center user and the reflection and transmission beamformings at the STAR-RIS, the total transmit power is minimized. Owing to the non-convexity caused by the coupling of active and passive beamformings, we first decouple it into two sub-problems. Then, the semidefinite relaxation and successive convex approximation are utilized to resolve them, respectively. Finally, we propose an alternating optimization algorithm to tackle them iteratively. Simulation results show that the proposed scheme can effectively reduce the transmit power while expanding the coverage.