The scrap steel fibers (SSF) produced by mechanical processing are processed and added to rubber concrete (RC) to make scrap steel fiber rubber concrete (SSFRC), so as to enhance the strength of the rubber concrete. SSFRC with a concrete strength grade of C30, a rubber dosage of 10%, and SSF dosages of 0, 0.5, 1.0, 1.5, 2.0, and 2.5% were prepared. Cubic compressive tests and drop hammer impact tests were conducted respectively on cubic and cylindrical specimens, and the influence of different weights of the drop hammer, different heights of the drop hammer, and different dosages of SSF on the results was analyzed. The results show that adding SSF to RC can effectively improve the compressive performance and impact resistance. When the fiber dosage is 1%, the compressive performance reaches the optimal level. With the increase of the weight and height of the drop hammer, the impact resistance of the specimens decreases significantly. With the increase of the dosage of SSF, the impact resistance of the specimens significantly improves. When the dosage of SSF is 1.5%, the impact resistance performance is the best. The finite element model was established using ABAQUS, and the simulation results were compared with the test results. The error was within an acceptable range.