Protecting personal data against exploitation of machine learning models is crucial. Recently, availability attacks have shown great promise to provide an extra layer of protection against the unauthorized use of data to train neural networks. These methods aim to add imperceptible noise to clean data so that the neural networks cannot extract meaningful patterns from the protected data, claiming that they can make personal data "unexploitable." This paper provides a strong countermeasure against such approaches, showing that unexploitable data might only be an illusion. In particular, we leverage the power of diffusion models and show that a carefully designed denoising process can counteract the effectiveness of the data-protecting perturbations. We rigorously analyze our algorithm, and theoretically prove that the amount of required denoising is directly related to the magnitude of the data-protecting perturbations. Our approach, called AVATAR, delivers state-of-the-art performance against a suite of recent availability attacks in various scenarios, outperforming adversarial training even under distribution mismatch between the diffusion model and the protected data. Our findings call for more research into making personal data unexploitable, showing that this goal is far from over. Our implementation is available at this repository: https://github.com/hmdolatabadi/AVATAR.