Composite asphalt binder has emerged as a potential solution for improving asphalt functionality at a wide spectrum of temperatures. Storage stability of modified binder remains a main concern to ensure homogeneity during various stages including its storage, pumping, transportation, and construction. The aim of this study was to assess the storage stability of composite asphalt binders fabricated using non-tire waste ethylene-propylene-diene-monomer (EPDM) rubber and waste plastic pyrolytic oil (PPO). The influence of addition of a crosslinking additive (sulfur) was also investigated. Two different approaches were employed in the fabrication of composite rubberized binders: (1) sequential introduction of PPO and rubber granules, and (2) inclusion of rubber granules pre-swelled with PPO at 90°C to the conventional binder. Based on the modified binder fabrication approaches and the addition of sulfur, four categories of modified binders were prepared, namely sequential (SA), sequential with sulfur (SA-S), pre-swelled (PA), and pre-swelled with sulfur (PA-S). For variable modifier dosages (EPDM:16%, PPO: 2, 4, 6, and 8%, and sulfur: 0.3%), a total of 17 combinations of rubberized asphalt were subjected to two durations of thermal storage (48 and 96 hours) and then characterized for their storage stability performance through various separation indices (SIs) based on conventional, chemical, microstructural, and rheological analyses. The optimal storage stability performance was achieved at a PPO dosage of 6% under the four candidate approaches. It was also observed that the SIs based on chemical analysis and rubber extraction test had a good correlation with rheology-based SIs compared to the conventionally used softening point difference. A composite modified binder with PPO and EPDM rubber having adequate storage stability is a promising step in the use of sustainable composite-modified binders in asphalt pavement construction.