Safe and efficient robotic navigation among humans is essential for integrating robots into everyday environments. Most existing approaches focus on simplified 2D crowd navigation and fail to account for the full complexity of human body dynamics beyond root motion. We present HumanMPC, a Model Predictive Control (MPC) framework for 3D Micro Air Vehicle (MAV) navigation among humans that combines theoretical safety guarantees with data-driven models for realistic human motion forecasting. Our approach introduces a novel twist to reachability-based safety formulation that constrains only the initial control input for safety while modeling its effects over the entire planning horizon, enabling safe yet efficient navigation. We validate HumanMPC in both simulated experiments using real human trajectories and in the real-world, demonstrating its effectiveness across tasks ranging from goal-directed navigation to visual servoing for human tracking. While we apply our method to MAVs in this work, it is generic and can be adapted by other platforms. Our results show that the method ensures safety without excessive conservatism and outperforms baseline approaches in both efficiency and reliability.