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It is inevitably crucial to classify sleep stage for the diagnosis of various diseases. However, existing automated diagnosis methods mostly adopt the "gold-standard" lectroencephalogram (EEG) or other uni-modal sensing signal of the PolySomnoGraphy (PSG) machine in hospital, that are expensive, importable and therefore unsuitable for point-of-care monitoring at home. To enable the sleep stage monitoring at home, in this paper, we analyze the relationship between infrared videos and the EEG signal and propose a new task: to classify the sleep stage using infrared videos by distilling useful knowledge from EEG signals to the visual ones. To establish a solid cross-modal benchmark for this application, we develop a new dataset termed as Seeing your Sleep Stage via Infrared Video and EEG ($S^3VE$). $S^3VE$ is a large-scale dataset including synchronized infrared video and EEG signal for sleep stage classification, including 105 subjects and 154,573 video clips that is more than 1100 hours long. Our contributions are not limited to datasets but also about a novel cross-modal distillation baseline model namely the structure-aware contrastive distillation (SACD) to distill the EEG knowledge to infrared video features. The SACD achieved the state-of-the-art performances on both our $S^3VE$ and the existing cross-modal distillation benchmark. Both the benchmark and the baseline methods will be released to the community. We expect to raise more attentions and promote more developments in the sleep stage classification and more importantly the cross-modal distillation from clinical signal/media to the conventional media.