Elastic moduli were theoretically computed using the Makishima–Mackenzie model for SiO2–Na2O–CaO glasses doped with Sb2O3 contents. The calculated elastic moduli (Young’s, bulk, shear, and longitudinal modulus) were observed to increase with an increase in the Sb2O3 contents. The microhardness showed an increase, while Poisson’s ratio decreased with the rise of the Sb2O3 contents. In addition, gamma-ray and neutron shielding parameters were evaluated for the investigated glasses. The linear attenuation coefficient (LAC) was simulated using the Monte Carlo N-particle transport code (MCNP-5). Other parameters, such as the mass attenuation coefficient (MAC), transmission factor (TF), and half-value layer, were calculated based on the simulated LAC. The addition of Sb2O3 content was observed to enhance the investigated glasses’ shielding parameters, where the highest LAC was achieved for the SCNSb10 glass with 10 mol% Sb2O3 and decreased from 0.441 to 0.154 cm−1 at gamma energies between 0.248 and 1.406 MeV. Furthermore, the fast neutron effective removal cross-section (∑R) was computed theoretically. The calculated results showed that the highest ∑R was equal to 0.0341 cm2g−1 and was obtained for the SCNSb0 glass, which had no Sb2O3 content, while the lowest ∑R was equal to 0.0286 cm2 g−1 for the SCNSb10 glass sample. The present work was carried out to examine the advantages of the soda–lime glasses with different Sb2O3 contents in several photon shielding applications, especially for radiation safety in nuclear installations.