For topologically nontrivial and very narrow bands, Coulomb repulsion between electrons has been predicted to give rise to a spontaneous fractional quantum-Hall (FQH) state in the absence of magnetic fields. Here we show that strongly correlated electrons in a t(2g)-orbital system on a triangular lattice self-organize into a spin-chiral magnetic ordering pattern that induces precisely the required topologically nontrivial and flat bands. This behavior is very robust and does not rely on fine-tuning. In order to go beyond mean field and to study the impact of longer-range interactions, we map the low-energy electronic states onto an effective one-band model. Exact diagonalization is then used to establish signatures of a spontaneous FQH state.