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A foundational theorem due to Buchdahl states that, within General Relativity (GR), the maximum compactness $\mathcal{C}\equiv GM/(Rc^2)$ of a static, spherically symmetric, perfect fluid object of mass $M$ and radius $R$ is $\mathcal{C}=4/9$. As a corollary, there exists a compactness gap between perfect fluid stars and black holes (where $\mathcal{C}=1/2$). Here we generalize Buchdahl's result by introducing the most general equation of state for elastic matter with constant longitudinal wave speeds and apply it to compute the maximum compactness of regular, self-gravitating objects in GR. We show that: (i) the maximum compactness grows monotonically with the longitudinal wave speed; (ii) elastic matter can exceed Buchdahl's bound and reach the black hole compactness $\mathcal{C}=1/2$ continuously; (iii) however, imposing subluminal wave propagation lowers the maximum compactness bound to $\mathcal{C}\approx0.462$, which we conjecture to be the maximum compactness of \emph{any} static elastic object satisfying causality; (iv) imposing also radial stability further decreases the maximum compactness to $\mathcal{C}\approx 0.389$. Therefore, although anisotropies are often invoked as a mechanism for supporting horizonless ultracompact objects, we argue that the black hole compactness cannot be reached with physically reasonable matter within GR and that true black hole mimickers require either exotic matter or beyond-GR effects.