Search and rescue operations require mobile robots to navigate unstructured indoor and outdoor environments. In particular, actively stabilized multirotor drones need precise movement data to balance and avoid obstacles. Combining radial velocities from on-chip radar with MEMS inertial sensing has proven to provide robust, lightweight, and consistent state estimation, even in visually or geometrically degraded environments. Statistical tests robustify these estimators against radar outliers. However, available work with binary outlier filters lacks adaptability to various hardware setups and environments. Other work has predominantly been tested in handheld static environments or automotive contexts. This work introduces a robust baro-radar-inertial odometry (BRIO) m-estimator for quadcopter flights in typical GNSS-denied scenarios. Extensive real-world closed-loop flights in cities and forests demonstrate robustness to moving objects and ghost targets, maintaining a consistent performance with 0.5 % to 3.2 % drift per distance traveled. Benchmarks on public datasets validate the system's generalizability. The code, dataset, and video are available at https://github.com/ethz-asl/rio.