  We study a doubly-degenerate orbital model on a honeycomb attice. This is a
model for orbital states in multiferroic layered iron oxides. The classical and
quantum models are analyzed by spin-wave approximation, Monte-Carlo simulation
and Lanczos method. A macroscopic number of degeneracy exists in the classical
ground state. In the classical model, a peak in the specific heat appears at a
temperature which is much lower than the mean-field ordering one. Below this
temperature, angle of orbital pseudo-spin is fixed, but conventional orbital
orders are not suggested. The degeneracy in the ground state is partially
lifted by thermal fluctuation. We suggest a role of zero-dimensional
fluctuation in hexagons on a low-temperature orbital structure. Lifting of the
degeneracy also occurs at zero temperature due to the quantum zero-point
fluctuation. We show that the ground-state wave function is well represented by
a linear combination of the states where a honeycomb lattice is covered by
nearest-neighboring pairs of orbitals with the minimum bond energy.
