  MV$_2$Ga$_4$ (M = Sc, Zr, Hf) compounds belong to an emerging class of
materials showing a unique combination of unusual superconducting behavior with
extended linear chains in the crystal structure. In order to gain insights
{into} its mechanical and thermal properties, we have performed
first-principles electronic-structure calculations in the framework of the
Density Functional Theory (DFT). From the calculated second-order elastic
constants, we have systematically shown that the extended linear vanadium chain
substructures indeed give rise to an anisotropic regime in the elastic and
mechanical moduli. The high density of valence and conduction electrons along
the linear vanadium chains leads to a directional dependence of the reciprocal
linear compressibility, Young's modulus and shear modulus. Poisson's ratio for
several elongation directions is also drastically affected by the presence of
extended V chains. If the elongation is along the V chains, all compounds
exhibit {practically} the same Poisson ratio in directions perpendicular to it,
further highlighting the importance of the V chains to the mechanical
properties. Moreover, based on our results, we have discussed the possible
consequences of the elastic anisotropy on the superconducting properties of the
compounds. Finally, using the Debye-Gr\"uneisen approximation, our calculations
of thermal properties show {a good agreement with the available experimental
low temperature heat capacity data above the superconducting critical
temperature.
