Although researchers have extensively investigated the dynamic mechanical properties of metallic materials at the macro-scale in terms of deformation mechanisms, strain rate strengthening and fracture mechanisms, they have rarely investigated the dynamic mechanical properties affected by size effects at the micro-/meso-scales. In this study, experiments on quasi-static compression and the split-Hopkinson pressure bar (SHPB) were conducted using oxygen-free high-conductivity (OFHC) copper with different geometrical and grain sizes to explore the size effects on various dynamic mechanical properties at the micro-/meso-scales. The experimental results showed that the quasi-static and dynamic mechanical properties of OFHC copper are affected by size effects at the micro-/meso-scales. In particular, OFHC copper was found to exhibit strain rate strengthening effects at the micro-/meso-scales, and the presence of micro-cracks was observed in the SHPB experimental specimens. In this study, after proposing a modified Johnson-Cook (J-C) constitutive model based on the surface layer model, we analysed the average relative error of the modified model and the original constitutive model. The influence of the size effect on the mechanical properties of materials is taken into account in the new modified constitutive model. The results of relative error analysis showed that the average relative errors for the original J-C model ranged from 3.2% to 18%, while the modified J-C model’s results ranged from 2.2% to 11%. The accuracy of the modified model is greatly improved. Following finite element analysis based on the modified J-C model and the original model, the results showed that the modified J-C model agreed well with the experimental results.