Hybrid nanostructures, in which a known number of quantum emitters are strongly coupled to a plasmonic resonator, should feature optical properties at room temperature such as few-photon nonlinearities or coherent superradiant emission. We demonstrate here that this coupling regime can only be reached with dimers of gold nanoparticles in stringent experimental conditions, when the interparticle spacing falls below 2 nm. Using a short transverse DNA double-strand, we introduce 5 dye molecules in the gap between two 40 nm gold particles and actively decrease its length down to sub-2 nm values by screening electrostatic repulsion between the particles at high ionic strengths. Single-nanostructure scattering spectroscopy then evidences the observation of a strong-coupling regime in excellent agreement with electrodynamic simulations. Furthermore, we highlight the influence of the planar facets of polycrystalline gold nanoparticles on the probability of observing strongly coupled hybrid nanostructures.