The decay psi(2S) --> etaJ/psi is used to measure, for the first time, all prominent eta-meson branching fractions with the same experiment in the same dataset, thereby providing a consistent treatment of systematics across branching fractions. We present results for eta decays to gamma gamma, pi(+)pi(-)pi(0), 3pi(0), pi(+)pi(-)gamma and e(+)e(-)gamma, accounting for 99.9% of all eta decays. The precision of several of the branching fractions and their ratios is improved. Two channels, pi(+)pi(-)gamma and e(+)e(-)gamma, show results that differ at the level of three standard deviations from those previously determined.