This study presents a generalized framework of vector calculus for non-integer dimensional spaces, motivated by the prevalence of fractals in nature. The work formulates first- and second-order differential operators, including gradient, divergence, and scalar and vector Laplacian, for scalar and rotationally covariant vector functions. This framework is applied to the thermoelastic response of an infinite fractal medium with a cylindrical cavity, a problem that incorporates thermoelastic mass diffusion and variable thermal conductivity through the Kirchhoff transformation. The system is analyzed under combined thermal and chemical shocks at the boundary, with the medium remaining mechanically fixed. The governing equations are solved using the Laplace transform method, and Zakian technique is employed for numerical inversion. The computational results indicate that parameters such as delay time and fractal dimension significantly influence the material's response. The graphical analysis visually examines the effects of different kernel functions, fractal dimension, variable thermal conductivity, nonlocal length and time scales on the thermoelastic response, providing a clear illustration of their impact. Specifically, an increase in fractal dimension leads to a more pronounced reduction in the thermoelastic response near the cylindrical cavity. Furthermore, an examination of different memory-dependent kernel functions reveals that nonlinear kernels demonstrate superior performance compared to linear kernels within this theoretical framework.