We have theoretically investigated the electric polarization and capacitance in a nano-scale coaxial-cylindrical capacitor made of a double-walled carbon nanotube, (6,0)@(36,0), using a first-principles density-functional method with the enforced Fermi-energy difference scheme. We show that the distribution of the accumulated charge in the inner tube is quantum-mechanically spilled outward, while that in the outer tube is penetrating inward. Reflecting these charge spills, the electrostatic capacitance of the system is larger than what would be expected from classical theory. Furthermore, the total capacitance of the system is smaller than the electrostatic capacitance, which is another quantum effect showing large contributions from the DOS of the electrodes to the capacitance. We believe that these two quantum effects in the capacitance are important for the successful design of carbon-nanotube devices.