We present visual inertial lidar legged navigation system (VILENS), an odometry system for legged robots based on factor graphs. The key novelty is the tight fusion of four different sensor modalities to achieve reliable operation when the individual sensors would otherwise produce degenerate estimation. To minimize leg odometry drift, we extend the robot's state with a linear velocity bias term, which is estimated online. This bias is observable because of the tight fusion of this preintegrated velocity factor with vision, lidar, and inertial measurement unit (IMU) factors. Extensive experimental validation on different ANYmal quadruped robots is presented, for a total duration of 2 h and 1.8 km traveled. The experiments involved dynamic locomotion over loose rocks, slopes, and mud, which caused challenges such as slippage and terrain deformation. Perceptual challenges included dark and dusty underground caverns, and open and feature-deprived areas. We show an average improvement of 62% translational and 51% rotational errors compared to a state-of-the-art loosely coupled approach. To demonstrate its robustness, VILENS was also integrated with a perceptive controller and a local path planner.