This paper proposed fully recycled aggregate concrete (FRAC), which was prepared by completely replacing natural aggregates (NA) with recycled coarse and fine aggregates (RCA and RFA) and partially replacing cement with recycled powder (RP). Four different aggregate systems (i.e., full-NA, full-RCA, full-RFA and full-RAs) and three different contents of the RP (i.e., 10%, 20% and 30% of the total binder) were taken as parameter variables. The mechanical properties (compressive and tensile strength) and uniaxial compressive stress–strain relationship of FRAC were investigated. The experimental results show that the addition of recycled materials has an adverse effect on the concrete strength. Incorporating RAs or RP has little effect on the ascending branch of the normalized stress–strain curves, but the former makes the descending branch of the curves steeper, while the latter makes them flatter. The elastic modulus is reduced due to the incorporation of RAs and RP. Based on the test results, an empirical stress–strain model was proposed and discussed.