A molecular network that exhibits critical correlations in the spatial order that is characteristic of a random, entropically stabilized, rhombus tiling is described. Specifically, we report a random tiling formed in a two-dimensional molecular network of p-terphenyl-3,5,3',5'-tetracarboxylic acid adsorbed on graphite. The network is stabilized by hexagonal junctions of three, four, five, or six molecules and may be mapped onto a rhombus tiling in which an ordered array of vertices is embedded within a nonperiodic framework with spatial fluctuations in a local order characteristic of an entropically stabilized phase. We identified a topological defect that can propagate through the network, giving rise to a local reordering of molecular tiles and thus to transitions between quasi-degenerate local minima of a complex energy landscape. We draw parallels between the molecular tiling and dynamically arrested systems, such as glasses.