Using the density functional theory and the embedded cluster model, we have calculated the energetics of carbon-vacancy complexes in Fe as a function of distance separating the carbon atom(s) and the vacancy along different crystallographic directions. Carbon is found to prefer off-center sites from the vacancy and maintains a constant distance of approximately 3.35 a0 from the nearest Fe atom(s) independent of its direction from the vacancy center. This distance agrees closely with that in stoichiometric Fe3C. Our results suggest that the equilibrium site of carbon is one where it is coordinated to three Fe atoms in its nearest neighbor shell. Further decoration of vacancies by more than one carbon atom has been found to be energetically favorable. The binding of carbon to iron atoms is not caused by charge transfer between the atoms, but rather has a magnetic origin. Positron lifetimes at vacancies and vacancy-carbon complexes have also been calculated for various configurations. Our results are in general agreement with experiment.