While robots deployed in real-world environments inevitably experience interaction failures, understanding how users respond through verbal and non-verbal behaviors remains under-explored in human-robot interaction (HRI). This gap is particularly significant in healthcare-inspired settings, where interaction failures can directly affect task performance and user trust. We present the Robot Failures in Medical HRI (RFM-HRI) Dataset, a multimodal dataset capturing dyadic interactions between humans and robots embodied in crash carts, where communication failures are systematically induced during item retrieval tasks. Through Wizard-of-Oz studies with 41 participants across laboratory and hospital settings, we recorded responses to four failure types (speech, timing, comprehension, and search) derived from three years of crash-cart robot interaction data. The dataset contains 214 interaction samples including facial action units, head pose, speech transcripts, and post-interaction self-reports. Our analysis shows that failures significantly degrade affective valence and reduce perceived control compared to successful interactions. Failures are strongly associated with confusion, annoyance, and frustration, while successful interactions are characterized by surprise, relief, and confidence in task completion. Emotional responses also evolve across repeated failures, with confusion decreasing and frustration increasing over time. This work contributes (1) a publicly available multimodal dataset (RFM-HRI), (2) analysis of user responses to different failure types and preferred recovery strategies, and (3) a crash-cart retrieval scenario enabling systematic comparison of recovery strategies with implications for safety-critical failure recovery. Our findings provide a foundation for failure detection and recovery methods in embodied HRI.