This letter investigates the bit error rate (BER) performance of simultaneous transmitting and reflecting reconfigurable intelligent surfaces (STAR-RISs) in non-orthogonal multiple access (NOMA) networks. In the investigated network, a STAR-RIS serves two non-orthogonal users located on either side of the surface by utilizing the mode switching protocol. We derive the closed-form and upper bound BER expressions in perfect and imperfect successive interference cancellation cases. Furthermore, asymptotic analyses are also conducted to provide further insights into the BER behavior in the high signal-to-noise ratio region. Finally, the accuracy of our theoretical analysis is validated through Monte Carlo simulations. The obtained results reveal that the BER performance of STAR-RIS-NOMA outperforms that of the classical NOMA system, and STAR-RIS might be a promising NOMA 2.0 solution.