The incorporation of crumb rubber (CR) into asphalt pavement materials can improve the performance of asphalt pavement and generate environmental benefits. However, the storage stability of the crumb rubber asphalt (CRA) remains an issue that needs to be resolved. This study explores the interaction laws among various modified materials based on the response surface methodology. Optimal preparation dosages of each material are determined, and performance predictions and validations are conducted. The storage stability of the CRA compounded with epoxidized soybean oil (ESO) and polyester fiber (PF) is investigated by combining traditional compatibility testing methods with refined characterization methods. The results indicate that the modification of CRA exhibits better rheological properties when the percentages of CR, PF, and ESO are 22%, 0.34%, and 3.21%, respectively. The addition of ESO effectively complements the light components of CRA to improve asphalt compatibility, and the addition of PF alleviates the adverse effects of ESO’s softening effect on rheological properties through stabilization and three-dimensional strengthening. The scientifically compounded additions of ESO and PF can effectively enhance the storage stability and rheological properties of CRA, promoting the development of sustainable and durable roads.