Hooke's law is derived in this answer by Taylor expanding an energy potential with arbitrary functional form. This is dependent on the displacements involved being "small".
By considering the scale of atomic forces in, say, a spring, how do we see this is true?
On the atomic scale there are two forces of interest. One is simple Coulombic attraction, and the other is a more complicated repulsive force which includes the Pauli exclusion principle and other quantum mechanical effects.
The attractive force can be represented by the potential: $$E_A=\frac{z_1 z_2 e^2}{4 \pi \epsilon_0 r}=-\frac{A}{r}$$
The repulsive force can be approximately represented by the potential: $$E_R=\frac{B}{r^n}$$ where $B$ and $n$ are parameters that depend on the atoms involved and capture the behavior of the QM effects. Often $n\approx 8$
This function, $E(r)=E_A+E_R$ has a minimum at $$r_0=\left( \frac{nB}{A}\right)^{\frac{1}{n-1}}$$ and around that point the potential is $$E(r)=E(r_0+\delta r)=-B(n-1)\left(\frac{nB}{A}\right)^{-\frac{n}{n-1}} + A(n-1)\left(\frac{nB}{A}\right)^{-\frac{3}{n-1}} \delta r^2 + O(\delta r^3)$$
Note that this has the form of the potential for a spring with the spring constant being obtained from the second term.
