  The incidence of multiplicity in cool, luminous massive stars is relatively
unknown compared to their hotter counterparts. Here we present radial velocity
(RV) measurements and investigate the multiplicity properties of red
supergiants (RSGs) in the 30~Doradus region of the Large Magellanic Cloud. We
provide absolute RV measurements for our sample and estimate line-of-sight
velocities for the Hodge 301 and SL 639 clusters, which agree well with those
of hot stars in the same clusters. By combining results for the RSGs with those
for nearby B-type stars, we estimate systemic velocities and velocity
dispersions for the two clusters, obtaining estimates for their dynamical
masses of $\log (M_{\rm dyn}/M_{\odot})=$3.8$\pm$0.3 for Hodge 301, and an
upper limit of $\log (M_{\rm dyn}/M_{\odot})$<3.1$\pm$0.8 for SL 639, assuming
Virial equilibrium. Analysis of the multi-epoch data reveals one RV-variable,
potential binary candidate (VFTS744), which is likely a semi-regular variable
asymptotic giant branch star. We estimate an upper limit on the observed binary
fraction for our sample of 0.3, where we are sensitive to maximum periods for
individual objects in the range of 1 to 10 000 days and mass-ratios above 0.3
depending on the data quality. From simulations of the RV measurements from
binary systems given the current data we conclude that systems within the
parameter range q>0.3, $\log$P[days]<3.5, would be detected by our variability
criteria, at the 90% confidence level. The intrinsic binary fraction,
accounting for observational biases, is estimated using simulations of binary
systems with an empirically defined distribution of parameters where orbital
periods are uniformly distributed in the 3.3<$\log$P[days]<4.3 range. A range
of intrinsic binary fractions are considered; a binary fraction of 0.3 is found
to best reproduce the observed data. [Abridged]
