The discovery of epsilon Indi Ba, Bb, a binary brown dwarf system very close to the Sun, makes possible a concerted campaign to characterise the physical parameters of two T dwarfs. Recent observations suggest substellar atmospheric and evolutionary models may be inconsistent with observations, but there have been few conclusive tests to date. We therefore aim to characterise these benchmark brown dwarfs to place constraints on such models. We have obtained high angular resolution optical, near-infrared, and thermal-infrared imaging and medium-resolution (up to R~5000) spectroscopy of epsilon Indi Ba, Bb with the ESO VLT and present VRIzJHKL'M' broad-band photometry and 0.63-5.1 micron spectroscopy of the individual components. The photometry and spectroscopy of the two partially blended sources were extracted with a custom algorithm. Furthermore, we use deep AO-imaging to place upper limits on the (model-dependent) mass of any further system members. We derive luminosities of log L/L_{sun}=-4.699+/-0.017 and -5.232+/-0.020 for epsilon Indi Ba, Bb, respectively, and using the dynamical system mass and COND03 evolutionary models predict a system age of 3.7-4.3Gyr, in excess of previous estimates and recent predictions from observations of these brown dwarfs. Moreover, the effective temperatures of 1352-1385K and 976-1011K predicted from the COND03 evolutionary models, for epsilon Indi Ba and Bb respectively, are in disagreement with those derived from the comparison of our data with the BT-Settl atmospheric models where we find effective temperatures of 1300-1340K and 880-940K, for epsilon Indi Ba and Bb respectively, with surface gravities of logg=5.25 and 5.50. Finally, we show that spectroscopically determined effective temperatures and surface gravities for ultra-cool dwarfs can lead to underestimated masses even where precise luminosity constraints are available.