In this study, a thermodynamic design of a two-stage cascade refrigeration system capable of reaching ultra-low temperatures was developed, and energy and exergy analyses were conducted. The performance of four refrigerant pairs—R744/R152a, R744/R32, R41/R152a, and R41/R32—was evaluated. Parametric analysis results, carried out across evaporator temperatures ranging from -70 °C to -50 °C and condenser temperatures between 25 °C and 45 °C, showed that R41/R152a was the most efficient pair. A maximum COP of 1.421 was achieved at -50 °C evaporator temperature, and the highest exergy efficiency of 0.4407 was recorded at -60 °C. In the second phase, an SVM approach was applied to predict COP and exergy efficiency, yielding very low MAE values of 0.0040 and 0.0009 in the test set, respectively. The outcomes of the energy-exergy analysis and SVM modeling are expected to provide valuable guidance for designing low-temperature cascade refrigeration systems.