How can a robot safely navigate around people exhibiting complex motion patterns? Reinforcement Learning (RL) or Deep RL (DRL) in simulation holds some promise, although much prior work relies on simulators that fail to precisely capture the nuances of real human motion. To address this gap, we propose Deep Residual Model Predictive Control (DR-MPC), a method to enable robots to quickly and safely perform DRL from real-world crowd navigation data. By blending MPC with model-free DRL, DR-MPC overcomes the traditional DRL challenges of large data requirements and unsafe initial behavior. DR-MPC is initialized with MPC-based path tracking, and gradually learns to interact more effectively with humans. To further accelerate learning, a safety component estimates when the robot encounters out-of-distribution states and guides it away from likely collisions. In simulation, we show that DR-MPC substantially outperforms prior work, including traditional DRL and residual DRL models. Real-world experiments show our approach successfully enables a robot to navigate a variety of crowded situations with few errors using less than 4 hours of training data.