Objective: Over the last two decades, radar-based contactless monitoring of vital signs (heartbeat and respiration rate) has raised increasing interest as an emerging and added value to health care. However, until now, the flaws caused by indoor multipath propagation formed a fundamental hurdle for the adoption of such technology in practical healthcare applications where reliability and robustness are crucial. Multipath reflections, originated from one person, combine with the direct signals and multipaths of other people and stationary objects, thus jeopardizing individual vital signs extraction and localization. This work focuses on tackling indoor multipath propagation. Methods: We describe a methodology, based on accurate models of the indoor multipaths and of the radar signals, that enables separating the undesired multipaths from desired signals of multiple individuals, removing a key obstacle to real-world contactless vital signs monitoring and localization. Results: We also demonstrated it by accurately measure individual heart rates, respiration rates, and absolute distances (range information) of paired volunteers in a challenging real-world office setting. Conclusion: The approach, validated using a frequency-modulated continuous wave (FMCW) radar, was shown to function in an indoor environment where radar signals are severely affected by multipath reflections. Significance: Practical applications arise for health care, assisted living, geriatric and quarantine medicine, rescue and security purposes.