Long-living mesoscopic clusters of a dense protein liquid are a necessary kinetic intermediate for the formation of solid aggregates of native and misfolded protein molecules; in turn, these aggregates underlie physiological and pathological processes and laboratory and industrial procedures. We argue that the clusters consist of a nonequilibrium mixture of single protein molecules and long-lived complexes of proteins. The puzzling mesoscopic size of the clusters is determined by the lifetime and diffusivity of these complexes. We predict and observe a crossover of cluster dynamics to critical-like density fluctuations at high protein concentrations. We predict and experimentally confirm that cluster dynamics obey a universal, diffusion-like scaling with time and wave vector, including in the critical-like regime. Nontrivial dependencies of the cluster size and volume fraction on the protein concentration are established. Possible mechanisms of complex formation include domain swapping, hydration forces, dispersive interactions, and other, system-specific, interactions. We highlight the significance of the hydration interaction and domain swapping with regard to the ubiquity of the clusters and their sensitivity to the chemical composition of the solvent. Our findings suggest novel ways to control protein aggregation.