As there is no specific boundary of an atom, how was Rutherford able to estimate the size of an atom? On the basis of the observations, Rutherford drew the following conclusions regarding the structure of an atom:

*

*Most of the space in the atom is empty as most of the  alpha particles passed through the foil
undeflected.


*A few positively charged alpha particles were deflected. The deflection must be due to enormous repulsive force showing that the positive charge of the atom is not spread throughout the atom as Thomson had presumed. The positive charge had to be concentrated in a very small volume that repelled and deflected the positively charged alpha particles.


*Calculations by Rutherford showed that the volume occupied by the nucleus was negligibly small compared to the total volume of the atom. The radius of the atom  is about $\mathbf{10^{–10}\,\mathrm{m}}$, while that of the nucleus is $\mathbf{10^{–15}\,\mathrm{m}}$
I know this model is unsatisfactory, but how did Rutherford calculate the radius of the atom to be $10^{-10}\,\mathrm{m}$?
 A: 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.
A: Rutherford probably estimated the size of gold atoms
as already sketched by @AndrewSteane in his comment.
The density of gold is $\rho=19.3\text{ g/cm}^3$.
The molar mass of gold was known from chemistry:
$m_\text{mol}= 197 \text{ g/mol}$.
From this you get the molar volume
$$V_\text{mol}=\frac{m_\text{mol}}{\rho}$$
Early estimations of Avogadro's constant (i.e. the number of atoms per mol)
were already known from physical experiments before Rutherford's time. Later experiments refined this value:
$$N_A=6.02\cdot 10^{23}\text{/mol}$$
Using this you get the volume per atom
$$V_\text{atom}=\frac{V_\text{mol}}{N_A}$$
Let us assume the gold atoms form a cubic lattice
(this is wrong, but good enough for an estimation).
Then each atom occupies a cube of edge length
$$d=\sqrt[3]{V_\text{atom}}$$
Doing the calculation we get
$$\begin{align}
d&=\sqrt[3]{V_\text{atom}}
=\sqrt[3]{\frac{V_\text{mol}}{N_A}}
=\sqrt[3]{{\frac{m_\text{mol}}{\rho\ N_A}}} \\
&=\sqrt[3]{{\frac{197 \text{ g/mol}}{19.3\text{ g/cm}^3 \cdot 6.02\cdot 10^{23}\text{/mol}}}} \\
&=\sqrt[3]{1.70\cdot 10^{-23}\text{cm}^3}
=\sqrt[3]{1.70\cdot 10^{-29}\text{m}^3}
=2.6\cdot 10^{-10}\text{ m}
\end{align}$$
And the radius of an atom is half of this cube edge length
$$r=\frac{d}{2}=1.3\cdot 10^{-10}\text{ m}$$
A: The size of the atom was estimated before Rutherford did his alpha particle experiment.
One way is to take a drop of oil of known radius and put in on water. It spreads out, given some time, to a large circle, of a small thickness.
From formulae for volume of a sphere and cylinder, the thickness can be calculated, if it's assumed to be one atom thick. That method calculates the size of an atom, originally done by Lord Rayleigh in about 1890, before Rutherford's experiment in 1908.
See also the grey box half way down this.
The value obtained was $1.6\times 10^{-9}$m.  Other scientists continued this work and refined the value.
