Well, high-tech cavities reach ~30MeV/m for singly charged particles. If you accelerate protons (which is a bad idea for current ion drives, but probably unimportant for relativistic speeds), this would need ~10m for some significant fraction of c. This does not account for the ion source, the issue that the cavities need pulses and not constant beams, and other problems. In addition, the fission reactor needs space, mass, a cooling cycle and large radiators.
Let's see: Assume a ship with a mass of 10 tons (probably too low, but whatever), about 10.8km/s (~10h to the moon) and ~1 hour acceleration/deceleration at the moon. There, I neglect its gravity and details of orbits - which is a good approximation at the moon (but not at earth). This requires an acceleration of 3m/s and therefore a thrust of 30kN.
With E=300MeV per proton, the momentum is ~800MeV/c, which requires 7*10^22 protons per second and the nice power of ~3300 GW, which is a bit more than the total output of all power plants on Earth ;).
However, with non-relativistic speeds, the thrust to power ratio is better. There, 2P=Fv. With 1GW, v~100km/s and the journey would require some tons (~2 per direction) of reaction mass. With 100MW and 1/10 of the acceleration, you get the same numbers (apart from the time to reach the moon) and the power becomes more realistic. However, packing all this stuff in 10 tons of total ship mass is still unrealistic.