Deep inertial sequence learning has shown promising odometric resolution over model-based approaches for trajectory estimation in GPS-denied environments. However, existing neural inertial dead-reckoning frameworks are not suitable for real-time deployment on ultra-resource-constrained (URC) devices due to substantial memory, power, and compute bounds. Current deep inertial odometry techniques also suffer from gravity pollution, high-frequency inertial disturbances, varying sensor orientation, heading rate singularity, and failure in altitude estimation. In this paper, we introduce TinyOdom, a framework for training and deploying neural inertial models on URC hardware. TinyOdom exploits hardware and quantization-aware Bayesian neural architecture search (NAS) and a temporal convolutional network (TCN) backbone to train lightweight models targetted towards URC devices. In addition, we propose a magnetometer, physics, and velocity-centric sequence learning formulation robust to preceding inertial perturbations. We also expand 2D sequence learning to 3D using a model-free barometric g-h filter robust to inertial and environmental variations. We evaluate TinyOdom for a wide spectrum of inertial odometry applications and target hardware against competing methods. Specifically, we consider four applications: pedestrian, animal, aerial, and underwater vehicle dead-reckoning. Across different applications, TinyOdom reduces the size of neural inertial models by 31× to 134× with 2.5m to 12m error in 60 seconds, enabling the direct deployment of models on URC devices while still maintaining or exceeding the localization resolution over the state-of-the-art. The proposed barometric filter tracks altitude within ±0.1m and is robust to inertial disturbances and ambient dynamics. Finally, our ablation study shows that the introduced magnetometer, physics, and velocity-centric sequence learning formulation significantly improve localization performance even with notably lightweight models.