  Castor is a system of six stars in which the two brighter objects, Castor A
and B, revolve around each other every $\sim$450 yr and are both short-period
spectroscopic binaries. They are attended by the more distant Castor C, which
is also a binary. Here we report interferometric observations with the CHARA
array that spatially resolve the companions in Castor A and B for the first
time. We complement these observations with new radial velocity measurements of
A and B spanning 30 yr, with the Hipparcos intermediate data, and with existing
astrometric observations of the visual AB pair obtained over the past three
centuries. We perform a joint orbital solution to solve simultaneously for the
three-dimensional orbits of Castor A and B as well as the AB orbit. We find
that they are far from being coplanar: the orbit of A is nearly at right angles
(92 degrees) relative to the wide orbit, and that of B is inclined about 59
degrees compared to AB. We determine the dynamical masses of the four stars in
Castor A and B to a precision better than 1%. We also determine the radii of
the primary stars of both subsystems from their angular diameters measured with
CHARA, and use them together with stellar evolution models to infer an age for
the system of 290 Myr. The new knowledge of the orbits enables us to measure
the slow motion of Castor C as well, which may assist future studies of the
dynamical evolution of this remarkable sextuple system.
