Describe the experimental basis for believing that the nucleus occupies a very small fraction of the volume of the atom

The nucleus contains most of the mass of an atom (protons and neutrons). the electron clouds are what take up much of its volume. Electron orbitals are huge in comparison to the size of the nucleus, but contain very little mass.

Rutherford's gold foil experiment:

Rutherford had the gold foil experiment where he bombarded alpha particles on a thin sheet of gold foil. Alpha particles are positively charged particles. It was seen that most of the particles just passed through the foil. Only a small percentage of the particles were deflected by positive charge (the protons in the nucleus).

He concluded:

The atom is made up mainly of empty space, with a small but massive region of concentrated charge at the center.

The charge on this central region was determined to be positive, and was named the atomic nucleus.

And the empty space is considered to fill with electrons, occupying most of the volume.