Contrastive self-supervised learning based on point-wise comparisons has been widely studied for vision tasks. In the visual cortex of the brain, neuronal responses to distinct stimulus classes are organized into geometric structures known as neural manifolds. Accurate classification of stimuli can be achieved by effectively separating these manifolds, akin to solving a packing problem. We introduce Contrastive Learning As Manifold Packing (CLAMP), a self-supervised framework that recasts representation learning as a manifold packing problem. CLAMP introduces a loss function inspired by the potential energy of short-range repulsive particle systems, such as those encountered in the physics of simple liquids and jammed packings. In this framework, each class consists of sub-manifolds embedding multiple augmented views of a single image. The sizes and positions of the sub-manifolds are dynamically optimized by following the gradient of a packing loss. This approach yields interpretable dynamics in the embedding space that parallel jamming physics, and introduces geometrically meaningful hyperparameters within the loss function. Under the standard linear evaluation protocol, which freezes the backbone and trains only a linear classifier, CLAMP achieves competitive performance with state-of-the-art self-supervised models. Furthermore, our analysis reveals that neural manifolds corresponding to different categories emerge naturally and are effectively separated in the learned representation space, highlighting the potential of CLAMP to bridge insights from physics, neural science, and machine learning.