The experimental measurement of correlation functions and critical exponents in disordered systems is key to testing renormalization group (RG) predictions. We mechanically unzip single DNA hairpins with optical tweezers, an experimental realization of the diffusive motion of a particle in a one-dimensional random force field, known as the Sinai model. We measure the unzipping forces $F_w$ as a function of the trap position $w$ in equilibrium, and calculate the force correlator $\Delta_m(w)$, its amplitude, and correlation length, finding agreement with theoretical predictions. We study the universal scaling properties by varying the trap strength $m^2$. Fluctuations of the position of the base pair at the unzipping junction $u$ scale as $u \sim m^{-\zeta}$, with a roughness exponent $ \zeta=1.34\pm0.06$, in agreement with the analytical prediction $\zeta = \frac{4}{3}$. Our study provides a single-molecule test of the functional RG approach for disordered elastic systems in equilibrium.