The existence, stability properties, and dynamical evolution of localized spatiotemporal chaos are studied. We provide evidence of spatiotemporal chaotic localized structures in a liquid crystal light valve experiment with optical feedback. The observations are supported by numerical simulations of the Lifshitz model describing the system. This model exhibits coexistence between a uniform state and a spatiotemporal chaotic pattern, which emerge as the necessary ingredients to obtain localized spatiotemporal chaos. In addition, we have derived a simplified model that allows us to unveil the front interaction mechanism at the origin of the localized spatiotemporal chaotic structures.