A novel biodegradable poly(glycolic acid-co-propylene fumarate) (PGA-co-PPF) copolymer has been synthesized via ring-opening polymerization. Graphene oxide (GO) and hydroxyapatite nanorods have been incorporated into PGA-co-PPF through electrospinning to yield hybrid nanocomposite fibers, and their morphology, water uptake, biodegradability, cytotoxicity, and mechanical, thermal and antibacterial properties have been analyzed. The addition of GO improved the dispersion of the HA nanorods within the matrix, and led to the formation of thinner fibers. The simultaneous incorporation of both nanofillers significantly increased the water absorption, biodegradation rate and protein adsorption capability of PGA-co-PPF. The hybrids induced higher osteoblast cell vitality and alkaline phosphatase activity than the neat copolymer and binary nanocomposites with either HA or GO, and showed higher biocidal activity against Gram-positive S. aureus and Gram-negative E. coli bacteria. Furthermore, experimental results revealed a synergistic effect of the nanofillers on improving the copolymer biocompatibility, thermal stability, stiffness, strength and toughness, and the nanocomposite with 20 wt% HA and 5 wt% GO exhibited the best combination of properties. The development of multifunctional polymer nanocomposite fibers with good biodegradability, very low toxicity, high tensile modulus and strong bactericidal activity opens up new perspectives for bone tissue engineering applications.