We present the results of a twofold experimental and computational study of (0001) GaN/AlN multilayers forming pseudomorphic superlattices. High-Resolution Transmission Electron Microscopy (HRTEM) shows that heterostructures with four c-lattice parameters thick GaN Quantum Wells (QW) are misfit-dislocation free. Accurate structural data are extracted from HRTEM images via a new methodology optimizing the residual elastic energy stored in the samples. Total energy calculations are performed with several models analogous to the experimental QWs with increasing thicknesses of GaN, whereas this of the AlN barrier is kept fixed at n = 8 c-lattice parameters. With vanishing external stresses, minimum energy configurations of the studied systems correspond to different strain states. Linear elasticity accurately yields the corresponding lattice parameters, suppressing the need for on-purpose total energy calculations. Theoretically justified parabolic fits of the excess interfacial energy yield the values of interfacial stress and elastic stiffness as functions of the GaN QW thickness. Total species-projected densities of states and gap values extracted from there allow deciphering the effect of the evolving strain on the electronic structure of the superlattice. It is found that the gap energy decreases linearly with increasing the strain of the QW. These results are briefly discussed in the light shed by previous works from the literature.