This study investigates the feasibility of collaborative use of recycled carbon fiber reinforced polymer (RCFRP) fibers and recycled aggregate (RA) in concrete, which is called RCFRP fiber reinforced RA concrete (RFRAC). The mechanical properties of the composite were studied through experimental investigation, considering different RCFRP fiber contents (0%, 0.5%, 1.0%, and 1.5% by volume) and different RA replacement rates (0%, 10%, 20%, and 30% by volume). Specifically, ten different mixes were designed to explore the flowability and compressive and flexural strengths of the proposed composite. Experimental results indicated that the addition of RCFRP fibers and RA had a relatively small influence on the compressive strength of concrete (less than 5%). Moreover, the addition of RA slightly decreased the flexural strength of concrete, while the addition of RCFRP fibers could significantly improve the flexural performance. For example, the flexural strength of RA concrete with 1.5% RCFRP fiber addition increased by 32.7%. Considering the good flexural properties of the composite and its potential in reducing waste CFRP and construction solid waste, the proposed RFRAC is promising for use in civil concrete structures with high flexural performance requirements.