The high flexibility, impermeability and strength of graphene membranes are key properties that can enable the next generation of nanomechanical sensors. However, for capacitive pressure sensors, the sensitivity offered by a single suspended graphene membrane is too small to compete with commercial sensors. Here, we realize highly sensitive capacitive pressure sensors consisting of arrays of nearly ten thousand small, freestanding double-layer graphene membranes. We fabricate large arrays of small-diameter membranes using a procedure that maintains the superior material and mechanical properties of graphene, even after high-temperature annealing. These sensors are readout using a low-cost battery-powered circuit board, with a responsivity of up to 47.8 aF Pa⁻¹ mm⁻², thereby outperforming the commercial sensors.