The presence of short-lived radionuclides (t 1/2 < 10 Myr) in the early solar system provides important information about the astrophysical environment in which the solar system formed. The discovery of now extinct 10Be (t 1/2 = 1.4 Myr) in calcium-aluminum-rich inclusions (CAIs) with Fractionation and Unidentified Nuclear isotope anomalies (FUN-CAIs) suggests that a baseline concentration of 10Be in the early solar system was inherited from the protosolar molecular cloud. In this paper, we investigate various astrophysical contexts for the nonthermal nucleosynthesis of 10Be by cosmic-ray-induced reactions. We first show that the 10Be recorded in FUN-CAIs cannot have been produced in situ by irradiation of the FUN-CAIs themselves. We then show that trapping of Galactic cosmic rays (GCRs) in the collapsing presolar cloud core induced a negligible 10Be contamination of the protosolar nebula, the inferred 10Be/9Be ratio being at least 40 times lower than that recorded in FUN-CAIs (10Be/9Be ~ 3 × 10–4). Irradiation of the presolar molecular cloud by background GCRs produced a steady-state 10Be/9Be ratio lsim 1.3 × 10–4 at the time of the solar system formation, which suggests that the presolar cloud was irradiated by an additional source of CRs. Considering a detailed model for CR acceleration in a supernova remnant (SNR), we find that the 10Be abundance recorded in FUN-CAIs can be explained within two alternative scenarios: (1) the irradiation of a giant molecular cloud by CRs produced by gsim 50 supernovae exploding in a superbubble of hot gas generated by a large star cluster of at least 20,000 members, and (2) the irradiation of the presolar molecular cloud by freshly accelerated CRs escaped from an isolated SNR at the end of the Sedov-Taylor phase. In the second picture, the SNR resulted from the explosion of a massive star that ran away from its parent OB association, expanded during most of its adiabatic phase in an intercloud medium of density of about 1 H-atom cm–3, and eventually interacted with the presolar molecular cloud only during the radiative stage. This model naturally provides an explanation for the injection of other short-lived radionuclides of stellar origin into the cold presolar molecular cloud (26Al, 41Ca, and 36Cl) and is in agreement with the solar system originating from the collapse of a molecular cloud shocked by a supernova blast wave.