Many intracellular membraneless organelles form via phase separation of intrinsically disordered proteins (IDPs) or regions (IDRs). These include the Caenorhabditis elegans protein LAF-1, which forms P granule-like droplets in vitro. However, the role of protein disorder in phase separation and the macromolecular organization within droplets remain elusive. Here, we utilize a novel technique, ultrafast-scanning fluorescence correlation spectroscopy, to measure the molecular interactions and full coexistence curves (binodals), which quantify the protein concentration within LAF-1 droplets. The binodals of LAF-1 and its IDR display a number of unusual features, including 'high concentration' binodal arms that correspond to remarkably dilute droplets. We find that LAF-1 and other in vitro and intracellular droplets are characterized by an effective mesh size of ∼3-8 nm, which determines the size scale at which droplet properties impact molecular diffusion and permeability. These findings reveal how specific IDPs can phase separate to form permeable, low-density (semi-dilute) liquids, whose structural features are likely to strongly impact biological function.