Sensory earables have evolved from basic audio enhancement devices into sophisticated platforms for clinical-grade health monitoring and wellbeing management. This paper introduces OmniBuds, an advanced sensory earable platform integrating multiple biosensors and onboard computation powered by a machine learning accelerator, all within a real-time operating system (RTOS). The platform's dual-ear symmetric design, equipped with precisely positioned kinetic, acoustic, optical, and thermal sensors, enables highly accurate and real-time physiological assessments. Unlike conventional earables that rely on external data processing, OmniBuds leverage real-time onboard computation to significantly enhance system efficiency, reduce latency, and safeguard privacy by processing data locally. This capability includes executing complex machine learning models directly on the device. We provide a comprehensive analysis of OmniBuds' design, hardware and software architecture demonstrating its capacity for multi-functional applications, accurate and robust tracking of physiological parameters, and advanced human-computer interaction.