Recently, deep neural networks have achieved excellent performance on low-light raw video enhancement. However, they often come with high computational complexity and large memory costs, which hinder their applications on resource-limited devices. In this paper, we explore the feasibility of applying the extremely compact binary neural network (BNN) to low-light raw video enhancement. Nevertheless, there are two main issues with binarizing video enhancement models. One is how to fuse the temporal information to improve low-light denoising without complex modules. The other is how to narrow the performance gap between binary convolutions with the full precision ones. To address the first issue, we introduce a spatial-temporal shift operation, which is easy-to-binarize and effective. The temporal shift efficiently aggregates the features of neighbor frames and the spatial shift handles the misalignment caused by the large motion in videos. For the second issue, we present a distribution-aware binary convolution, which captures the distribution characteristics of real-valued input and incorporates them into plain binary convolutions to alleviate the degradation in performance. Extensive quantitative and qualitative experiments have shown our high-efficiency binarized low-light raw video enhancement method can attain a promising performance. The code is available at https://github.com/zhanggengchen/BRVE.