The quantum breakdown model is a model of randomly interacting fermions, motivated by the physics of dielectric breakdown. Here, we establish classification of symmetry and quantum chaos in a zero-dimensional, all-to-all-interacting version of the quantum breakdown model. It exhibits the
Z4 periodicity with respect to the number of fermionic modes, reminiscent of the symmetry classification of the Sachdev-Ye-Kitaev model. A unique feature of the quantum breakdown model is that it realizes all the five classes with chiral symmetry and hosts an exponentially large number of many-body zero modes protected by the chiral index. We elucidate the separation and scaling of the spectral gap and demonstrate the symmetry-enriched hard-edge spectral statistics as signatures of quantum chaos in the chiral symmetry classes.