  Earlier measurements of the masses and radii of the detached eclipsing binary
V20 in the open cluster NGC 6791 were accurate enough to demonstrate that there
are significant differences between current stellar models. Here we improve on
those results and add measurements of two additional detached eclipsing
binaries, the cluster members V18 and V80. The enlarged sample sets much
tighter constraints on the properties of stellar models than has hitherto been
possible, thereby improving both the accuracy and precision of the cluster age.
We employed (i) high-resolution UVES spectroscopy of V18, V20 and V80 to
determine their spectroscopic effective temperatures, [Fe/H] values, and
spectroscopic orbital elements, and (ii) time-series photometry from the Nordic
Optical Telescope to obtain the photometric elements. The masses and radii of
the V18 and V20 components are found to high accuracy, with errors on the
masses in the range 0.27-0.36% and errors on the radii in the range 0.61-0.92%.
V80 is found to be magnetically active, and more observations are needed to
determine its parameters accurately. The metallicity of NGC 6791 is measured
from disentangled spectra of the binaries and a few single stars to be [Fe/H]=
+0.29 \pm 0.03 (random) \pm 0.07 (systematic). The cluster reddening and
apparent distance modulus are found to be E(B - V) = 0.160 \pm 0.025 and (m -
M)V = 13.51 \pm 0.06 . A first model comparison shows that we can constrain the
helium content of the NGC 6791 stars, and thus reach a more accurate age than
previously possible. It may be possible to constrain additional parameters, in
particular the C, N, and O abundances. This will be investigated in paper II.
