Effective control of terahertz radiation requires fast and efficient modulators with a large modulation depth-a challenge that is often tackled by using metamaterials. Metamaterial-based active modulators can be created by placing graphene as a tuneable element shunting regions of high electric field confinement in metamaterials. However, in this common approach, the graphene is used as a variable resistor, and the modulation is achieved by resistive damping of the resonance. In combination with the finite conductivity of graphene due to its gapless nature, achieving 100% modulation depth using this approach remains challenging. Here, we embed nanoscale graphene capacitors within the gaps of the metamaterial resonators, and thus switch from a resistive damping to a capacitive tuning of the resonance. We further expand the optical modulation range by device excitation from its substrate side. As a result, we demonstrate terahertz modulators with over four orders of magnitude modulation depth (45.7 dB at 1.68 THz and 40.1 dB at 2.15 THz), and a reconfiguration speed of 30 MHz. These tuneable capacitance modulators are electrically controlled solid-state devices enabling unity modulation with graphene conductivities below 0.7 mS. The demonstrated approach can be applied to enhance modulation performance of any metamaterial-based modulator with a 2D electron gas. Our results open up new frontiers in the area of terahertz communications, real-time imaging, and wave-optical analogue computing.