Is there a way to experimentally measure entropy?
The most common way of measuring $\Delta S^\circ$ for a chemical reaction is probably by making a van't Hoff plot. You measure the equilibrium constant $K$ at different temperatures and plot $\ln K$ vs $T^{-1}$.  The $y$-intercept = $R\Delta S^\circ$ and the slope = $-R\Delta H^\circ$.

Another option is to measure $\Delta H^\circ$ by calorimetry and measure $K$ by some other means. Then compute $\Delta G^\circ$ from $K$ and solve for $\Delta S^\circ$