The bodies of the Solar System exist in a variety of irregular shapes. Studies of those shapes are conducted to infer information about the internal composition, structure, and history of those bodies. However, such inferences require knowing how the composition and structure or history relates to the shape and internal forces. That connection is known only for fluid bodies, where the permissible equilibrium states were discovered centuries ago by Newton, Maclaurin, Jacobi, Poincaré, and Roche. While others have given results for linear elastic solid bodies, the elastic problem is not uniquely posed, since elastic solutions depend on an implicit assumption about the existence and shape of an initial stress-free state. The present states of Solar System bodies are a culmination of complicated past histories, possibly involving collisions, disruption, melting, accumulation, and large-scale yielding and reshaping. Such processes create underlying residual stress fields that cannot be known. Here I present an approach in the same spirit as for the fluid bodies: limits on equilibrium shapes are determined. Results are obtained for a cohesionless elastic-plastic solid with a Mohr-Coloumb yield criteria. That model is commonly used in soil mechanics and is appropriate for "rubble pile" reaccumulated asteroids that have negligible cohesive forces. It is possible to determine limit equilibrium stress fields and shapes that are independent of past histories, using the approaches of limit analyses of elastic-plastic theories. The results show that for these bodies there exists a region of permissible combinations of shape and spin rates, centered about the unique equilibrium fluid states of Maclaurin and Jacobi. The database on asteroids is compared to those equilibrium states. Few asteroids are outside the limit shape envelopes according to this theory. The application of the analysis to Phobos is also presented, assuming that the rubble-pile model is appropriate. The deformation that would occur as it moves closer to Mars is determined; it is shown to be unstable and globally catastrophic at about 2.1 Mars radii.