Bovine serum albumin (BSA) and poly(diallyldimethylammonium chloride) (PDADMAC) spontaneously form, over a range of ionic strength I and pH, dense fluids rich in both macroions. To study their nanostructure, these coacervates were prepared at low I and high pH (strong interaction) or at high I and lower pH (weaker interaction), with polymer MWs ranging from 90K to 700K, and then examined by dynamic light scattering (DLS) and rheology. DLS shows a dominant and surprisingly fast protein diffusional mode independent of polymer MW; accompanied by robust slow modes, slower by 1-2 orders of magnitude, which are also insensitive to MW and are present regardless of I, pH, and sample aging. High MW sensitivity was observed by rheology for the terminal time (order of milliseconds), which increased as well with the strength of polyelectrolyte-protein interaction. Viscoelastic behavior also indicated a tenuous network, solidlike at low strain but re-forming after breakage by shear. Two models, both of which have strengths and defects, are put forward: (I) macroion-rich domains dispersed in a continuum of macroion-poor domains near the percolation limit and (II) a semidilute solution of PDADMAC chains with interchain friction modulated by transient BSA-PDADMAC association.