Bound states in the continuum (BICs) are spatially localized modes embedded in the spectrum of extended states, typically stabilized by symmetry or interference. While extensively studied in single-particle and linear systems, the many-body regime of BICs remains largely unexplored. Here, we show that giant atoms, quantum emitters coupled nonlocally to structured waveguides, can host robust doublon BICs, i.e., two-photon bound states stabilized by destructive interference and interactions. We first analyze a driven two-photon emission process and show how doublon BICs arise and mediate decoherence-free interaction between distant atoms. We then show that these doublon BICs also emerge under natural, undriven dynamics via a virtual two-photon emission process in three-level giant atoms. Our results put forward giant atoms as a platform to realize two- and eventually many-body BICs, revealing an interference-based mechanism for stabilizing many-body localized states.