The exchange bias (EB) effect and an appearance of the Griffiths phase (GP) were found in half-doped cobaltite $\mathrm{G}{\mathrm{d}}_{0.5}\mathrm{S}{\mathrm{r}}_{0.5}\mathrm{Co}{\mathrm{O}}_{3\ensuremath{-}\ensuremath{\delta}}$ exhibiting a significant quenched disorder due to the ion size mismatch between Sr and Gd. The disorder weakens the ferromagnetic (FM) interactions between Co ions, leading to low Curie temperature ${T}_{C}=90\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ and to a highly nonhomogenous magnetic state above ${T}_{C}$. A clear GP behavior was detected in the temperature range between ${T}_{C}$ and the Griffiths temperature ${T}_{G}=225\phantom{\rule{0.16em}{0ex}}\mathrm{K}$. Moreover, this GP demonstrates a unique feature; namely, the appreciable EB effect exists concomitantly within the GP, suggesting coexisting FM and antiferromagnetic nanocluster phases. It was found that the EB exists in $\mathrm{G}{\mathrm{d}}_{0.5}\mathrm{S}{\mathrm{r}}_{0.5}\mathrm{Co}{\mathrm{O}}_{3\ensuremath{-}\ensuremath{\delta}}$ for the entire temperature range below ${T}_{G}=225\phantom{\rule{0.16em}{0ex}}\mathrm{K}$, in contrast to the limited low-temperature EB observed so far in perovskite cobaltites. The EB has a different nature in the FM cluster phase below ${T}_{C}$ and in the GP in temperature interval ${T}_{C}lTl{T}_{G}$. The cooling field effect on EB was examined in the GP, and the size of FM clusters was determined to be equal to 6.5 nm.