Layered cobaltates embody novel realizations of correlated matter on a spin-1/2 triangular lattice. We report a high-resolution systematic photoemission study of the insulating cobaltates. The observation of a single-particle gap opening and band folding provides direct evidence of anisotropic particle-hole instability on the Fermi surface due to its unique topology. Overlap of the measured Fermi surface is observed with the square root 3xsquare root 3 charge-order Brillouin zone near x=1/3 but not at x=1/2 where the insulating transition is actually observed. Unlike conventional density waves, charge stripes, or band insulators, the onset of the gap depends on the quasiparticle's quantum coherence which is found to occur well below the disorder-order symmetry breaking temperature of the crystal (the first known example of its kind).