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Motivated by the recent studies of quasi-periodic equilibria of dense packing of diblock and star polymers, this work introduces the Hookean-Voronoi energy, a minimal model for the packing of soft, deformable balls. Considering the planar case, we investigate the equilibrium packings of N deformable circles in a periodic square, deriving a reduced formulation of the system and establish that it has a large, O(N 2 ), family of ordered "single-string" minimizers. Through numerical investigation of gradient flows from random initial data, we show that for modest values of N the system frequently equilibrates to quasi-ordered states with low energy and large basins of attraction. For larger N the energy of equilibria approach a limiting distribution, shaped by two mechanisms: a proliferation of moderate-energy disordered equilibria that block access of the gradient flow to lower energy quasi-ordered states and a rigid threshold on the maximum energy of stable states that precludes high-energy equilibria. The system frustration, measuring gap between the average equilibrium energy of the gradient flow and the system ground state, increases with N but saturates.