A novel near-field transmission framework is proposed for dynamic metasurface antenna (DMA)-enabled non-orthogonal multiple access (NOMA) networks. The base station (BS) exploits the hybrid beamforming to communicate with multiple near users (NUs) and far users (FUs) using the NOMA principle. Based on this framework, two novel beamforming schemes are proposed. 1) For the case of the grouped users distributed in the same direction, a beam-steering scheme is developed. The metric of beam pattern error (BPE) is introduced for the characterization of the gap between the hybrid beamformers and the desired ideal beamformers, where a two-layer algorithm is proposed to minimize BPE by optimizing hybrid beamformers. Then, the optimal power allocation strategy is obtained to maximize the sum achievable rate of the network. 2) For the case of users randomly distributed, a beam-splitting scheme is proposed, where two sub-beamformers are extracted from the single beamformer to serve different users in the same group. An alternating optimization (AO) algorithm is proposed for hybrid beamformer optimization, and the optimal power allocation is also derived. Numerical results validate that: 1) the proposed beamforming schemes exhibit superior performance compared with the existing imperfect-resolution-based beamforming scheme; 2) the communication rate of the proposed transmission framework is sensitive to the imperfect distance knowledge of NUs but not to that of FUs.