Post-training algorithms based on deep reinforcement learning can push the limits of robotic models for specific objectives, such as generalizability, accuracy, and robustness. However, Intervention-requiring Failures (IR Failures) (e.g., a robot spilling water or breaking fragile glass) during real-world exploration happen inevitably, hindering the practical deployment of such a paradigm. To tackle this, we introduce Failure-Aware Offline-to-Online Reinforcement Learning (FARL), a new paradigm minimizing failures during real-world reinforcement learning. We create FailureBench, a benchmark that incorporates common failure scenarios requiring human intervention, and propose an algorithm that integrates a world-model-based safety critic and a recovery policy trained offline to prevent failures during online exploration. Extensive simulation and real-world experiments demonstrate the effectiveness of FARL in significantly reducing IR Failures while improving performance and generalization during online reinforcement learning post-training. FARL reduces IR Failures by 73.1% while elevating performance by 11.3% on average during real-world RL post-training. Videos and code are available at https://failure-aware-rl.github.io.