In network theory an increasing amount of research has focused on understanding how the degree distribution affects the structural and dynamical behaviors of the system. However, from Biology to real-world infrastructures, many relevant complex systems appear to be spatially embedded, while keeping the underlying complexity of their interaction topology. Despite many relevant advances obtained in studying the structure and function of complex networks, addressing the collective properties of spatially embedded systems has proved to be challenging, both from the computational and the theoretical view-point. Here we introduce a realistic model for a spatial network, constructed by randomly linking lattice sites with link-lengths following an exponential distribution with some given characteristic scale $\zeta$. We find that this network topology exhibits crossover phenomena from mean-field to classical lattice behaviors, both at and away from criticality. By means of extensive numerical simulations, we have studied how the main critical exponents characterizing percolation-based processes crosses over, smoothly changing from the mean-field universality class to the one for lattices in two-dimensions. Our results point out to a challenging direction, so far overlooked, in the study of the structure and function of spatially embedded complex systems.