Learning generalizable manipulation policies hinges on data, yet robot manipulation data is scarce and often entangled with specific embodiments, making both cross-task and cross-platform transfer difficult. We tackle this challenge with task-agnostic embodiment modeling, which learns embodiment dynamics directly from task-agnostic action data and decouples them from high-level policy learning. By focusing on exploring all feasible actions of the embodiment to capture what is physically feasible and consistent, task-agnostic data takes the form of independent image-action pairs with the potential to cover the entire embodiment workspace, unlike task-specific data, which is sequential and tied to concrete tasks. This data-driven perspective bypasses the limitations of traditional dynamics-based modeling and enables scalable reuse of action data across different tasks. Building on this principle, we introduce AnyPos, a unified pipeline that integrates large-scale automated task-agnostic exploration with robust embodiment modeling through inverse dynamics learning. AnyPos generates diverse yet safe trajectories at scale, then learns embodiment representations by decoupling arm and end-effector motions and employing a direction-aware decoder to stabilize predictions under distribution shift, which can be seamlessly coupled with diverse high-level policy models. In comparison to the standard baseline, AnyPos achieves a 51% improvement in test accuracy. On manipulation tasks such as operating a microwave, toasting bread, folding clothes, watering plants, and scrubbing plates, AnyPos raises success rates by 30-40% over strong baselines. These results highlight data-driven embodiment modeling as a practical route to overcoming data scarcity and achieving generalization across tasks and platforms in visuomotor control. Project page: https://embodiedfoundation.github.io/vidar_anypos.