Answer
See explanation
Work Step by Step
This claim is not reasonably accurate. The size of a nucleus is on the order of 10^-15 meters, while the average distance of an electron from the nucleus is on the order of 10^-10 meters.
To provide mathematical support:
The size of a grape is about 2.0 cm or 0.02 m. The electron distance is $0.02\times 10^5=2,000$m$\approx 1.243$ miles.
This means that, quantitatively, the claim is reasonably accurate: scaling a fm-scale nucleus to a grape (centimeter scale) puts the electron cloud on the order of about 1 mile is a perfectly acceptable, memorable approximation.
