We report individual dynamical masses of 66.92{+/-}0.36 MJup and 53.25{+/-}0.29MJup for the binary brown dwarfs {epsilon} Indi Ba and Bb, measured from long-term (~10yr) relative orbit monitoring and absolute astrometry monitoring data on the Very Large Telescope (VLT). Relative astrometry with NACO fully constrains the Keplerian orbit of the binary pair, while absolute astrometry with FORS2 measures the system's parallax and mass ratio. We find a parallax consistent with the Hipparcos and Gaia values for {epsilon} Indi A, and a mass ratio for {epsilon} Indi Ba to Bb precise to better than 0.2%. {epsilon} Indi Ba and Bb have spectral types T1-1.5 and T6, respectively. With an age of 3.5_-1.0_^+0.8^Gyr from {epsilon} Indi A's activity, these brown dwarfs provide some of the most precise benchmarks for substellar cooling models. Assuming coevality, the very different luminosities of the two brown dwarfs and our moderate mass ratio imply a steep mass-luminosity relationship (L{prop}M^5.37{+/-}0.08^) that can be explained by a slowed cooling rate in the L/T transition, as previously observed for other L/T binaries. Finally, we present a periodogram analysis of the near-infrared photometric data, but find no definitive evidence of periodic signals with a coherent phase.