Ultrasound videos are an important form of clinical imaging data, and deep learning-based automated analysis can improve diagnostic accuracy and clinical efficiency. However, the scarcity of labeled data and the inherent challenges of video analysis have impeded the advancement of related methods. In this work, we introduce E-ViM$^3$, a data-efficient Vision Mamba network that preserves the 3D structure of video data, enhancing long-range dependencies and inductive biases to better model space-time correlations. With our design of Enclosure Global Tokens (EGT), the model captures and aggregates global features more effectively than competing methods. To further improve data efficiency, we employ masked video modeling for self-supervised pre-training, with the proposed Spatial-Temporal Chained (STC) masking strategy designed to adapt to various video scenarios. Experiments demonstrate that E-ViM$^3$ performs as the state-of-the-art in two high-level semantic analysis tasks across four datasets of varying sizes: EchoNet-Dynamic, CAMUS, MICCAI-BUV, and WHBUS. Furthermore, our model achieves competitive performance with limited labels, highlighting its potential impact on real-world clinical applications.