Even if the electron cloud around an atom is diffuse, when packed together atoms take up a well-defined volume
The previous answers explain how the average volume taken up by an atom is calculated. And this is, indeed, what was done pre-Rutherford. This leaves the question of why the fuzzy region of space occupied by the electrons in an atom can be said to occupy a specific volume.
This needs an understanding of the forces involved when atoms come close together. Just because the isolated atom has a cloud of electrons which is "fuzzy" (at least in the sense that there is a small probability of finding an electron a long way away from the nucleus) this doesn't mean that two atoms interacting don't settle a definite (or, at least, fairly precise) distance apart.
That distance depends on the balance of attractive and repulsive forces among the atoms that are interacting. Some isolated atoms see strong forces when they come close (two isolated hydrogen atoms actually form a bond when they come close as energy can be released by sharing the electrons. This results in a bond with a very specific length. Crudely, the attractive forces of the bond counteract the repulsive forces driving the nuclei apart. But a quantum-mechanical calculation is needed to give a fuller picture taking into account things like the pauli exclusion principle).
A simpler situation arises when molecules or noble gas atoms not keen on making further bonds come into contact. Despite the "fuzzy" electron clouds they still see a mix of repulsive and attractive forces. The forces can be thought of as arising from quantum fluctuations in the electron clouds leading to very short lived dipoles that create short term forces pulling molecules or atoms together until the repulsive forces balance them out. The form of this overall potential is well understood (and can be derived from some fairly complex quantum calculations) but the details are not important. What matters is that atoms settle a fixed distance apart when the forces balance. Chemists tend to call this distance the atomic radius (or the Van der Walls radius after the name of the forces involved) and this is often considered the "size" of an atom. Many molecular solids are held together with these forces.
Other compounds have further types of bonding. Some solids, like diamond, are held together with an infinite array of strong covalent bonds. In these the atoms sit a specific distance apart caused by the length of the bond which, in turn is caused by the equilibrium of quantum forces pulling the atoms closer and others pushing them apart. Metals have many metal atoms sitting in a sea of free electrons holding them together against atomic repulsion.
The point, in all of these cases, is that what determines the definite and specific size of atoms in solids or molecules is a balance between repulsive and attractive forces. those forces reach an equilibrium at a fairly specific and definite point which can be used to define a fairly precise size for an atom despite the apparent "fuzziness" of the single atom's electron cloud.
IF you look at the forces involved in interacting atoms you get a much less fuzzy view of atomic size than you do by trying to draw an arbitrary boundary on electron density of the atom's electron cloud. That is how Rutherford could define the size of a gold atom.