We investigate the ground state of the irrationally frustrated Josephson junction array with a controlling anisotropy parameter lambda that is the ratio of the longitudinal Josephson coupling to the transverse one. We find that the ground state has one-dimensional periodicity whose reciprocal lattice vector depends on lambda and is incommensurate with the substrate lattice. Approaching the isotropic point lambda = 1, the so-called hull function of the ground state exhibits analyticity breaking similar to the Aubry transition in the Frenkel-Kontorova model. We find a scaling law for the harmonic spectrum of the hull functions, which suggests the existence of a characteristic length scale diverging at the isotropic point. This critical behavior is directly connected to the jamming transition previously observed in the current-voltage characteristics by a numerical simulation. On top of the ground state there is a gapless continuous band of metastable states, which exhibit the same critical behavior as the ground state.