Omnidirectional Depth Estimation has broad application prospects in fields such as robotic navigation and autonomous driving. In this paper, we propose a robotic prototype system and corresponding algorithm designed to validate omnidirectional depth estimation for navigation and obstacle avoidance in real-world scenarios for both robots and vehicles. The proposed HexaMODE system captures 360$^\circ$ depth maps using six surrounding arranged fisheye cameras. We introduce a combined spherical sweeping method and optimize the model architecture for proposed RtHexa-OmniMVS algorithm to achieve real-time omnidirectional depth estimation. To ensure high accuracy, robustness, and generalization in real-world environments, we employ a teacher-student self-training strategy, utilizing large-scale unlabeled real-world data for model training. The proposed algorithm demonstrates high accuracy in various complex real-world scenarios, both indoors and outdoors, achieving an inference speed of 15 fps on edge computing platforms.