In this article, we propose a novel simultaneous transmitting and reflecting reconfigurable intelligent surface (STAR-RIS)-assisted nonorthogonal multiple access (NOMA) system. Unlike most of the STAR-RIS-assisted NOMA works, we target scalable phase shift design for our proposed system that requires a reduced channel estimation overhead. Within this perspective, we propose novel algorithms to partition the STAR-RIS surface among the available users. These algorithms aim to determine the proper number of transmitting/reflecting elements needs to be assigned to each user in order to maximize the system sum rate while guaranteeing the quality-of-service requirements for individual users. For the proposed system, we derive closed-form analytical expressions for the outage probability (OP) and its corresponding asymptotic behavior under different user deployments. Finally, Monte Carlo simulations are performed in order to verify the correctness of the theoretical analysis. It is shown that the proposed system outperforms the classical NOMA and orthogonal multiple access systems in terms of OP and sum rate, under spatial correlation and phase errors.