By means of confocal laser scanning microscopy, we directly visualize and quantify the local spatial structure of dense colloidal gels of varying volume fraction formed by short-range attractive interparticle interactions. We find that the short-range structure of the gels is similar to that of dense amorphous hard-sphere systems. However, the long-range structure is characterized by voids and density correlations that differ remarkably from amorphous hard spheres and glasses formed through repulsive interactions. Quantification of the volume fraction dependence of the gel isothermal compressibility and Voronoi volume distribution establishes that long-range structural heterogeneity is a distinguishing feature of the static structure of dense colloidal gels formed by short-range attractions.