Sleep, as one of the key functions of the brain, is crucial for maintaining the physical and mental energy required for daily activities. Sleep staging classification is a common method for monitoring the quality of human sleep. Previous sleep classification methods were unable to effectively recognize the complex spatiotemporal features of EEG signals, leading to inadequate classification results. In this paper, we propose a Dual-Stream Spatiotemporal 3D Convolutional Neural Network (DST-3DCNN). Our method processes features in both the temporal and spatial streams through dual branches, learning the intrinsic connections between Electroencephalogram(EEG) channels to better analyze the spatiotemporal characteristics of signals. We also employ a multi-level residual network to fuse features from the temporal and spatial domains, enhancing the effectiveness of feature fusion. Moreover, to reduce the complexity of the network, we incorporate an efficient channel attention mechanism into DST-3DCNN, which enhances the perception of important features. Experimental results on two real-world datasets show that our model achieves an accuracy of 82.4% in sleep staging classification on ISRUC-S3, with an F1-Score of 0.816 and a kappa value of 0.763, demonstrating strong competitiveness. The accuracy on the ISRUC-S1 dataset is 82.0%, the F1-score is 0.794, and the Kappa value is 0.762.This study provides a new technical framework for automatic sleep staging and also offers a new approach for processing complex EEG signals.