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While classical approaches to autonomous robot navigation currently enable operation in certain environments, they break down in tightly constrained spaces, e.g., where the robot needs to engage in agile maneuvers to squeeze between obstacles. Recent machine learning techniques have the potential to address this shortcoming, but existing approaches require vast amounts of navigation experience for training, during which the robot must operate in close proximity to obstacles and risk collision. In this paper, we propose to side-step this requirement by introducing a new machine learning paradigm for autonomous navigation called learning from hallucination (LfH), which can use training data collected in completely safe environments to compute navigation controllers that result in fast, smooth, and safe navigation in highly constrained environments. Our experimental results show that the proposed LfH system outperforms three autonomous navigation baselines on a real robot and generalizes well to unseen environments, including those based on both classical and machine learning techniques.