  The quasi-skutterudite superconductor Sr$_3$Rh$_4$Sn$_{13}$ features a
pronounced anomaly in electrical resistivity at $T^*\sim$138 K. We show that
the anomaly is caused by a second-order structural transition, which can be
tuned to 0 K by applying physical pressure and chemical pressure via the
substitution of Ca for Sr. A broad superconducting dome is centred around the
structural quantum critical point. Detailed analysis of the tuning parameter
dependence of $T^*$ as well as insights from lattice dynamics calculations
strongly support the existence of a structural quantum critical point at
ambient pressure when the fraction of Ca is 0.9 (i.e., $x_c=0.9$). This
establishes (Ca$_x$Sr$_{1-x}$)$_3$Rh$_4$Sn$_{13}$ series as an important system
for exploring the physics of structural quantum criticality without the need of
applying high pressures.
