Strontium ruthenate (Sr₂RuO₄) has long been thought to host a spin-triplet chiral p-wave superconducting state. However, the singletlike response observed in recent spin-susceptibility measurements casts serious doubts on this pairing state. Together with the evidence for broken time-reversal symmetry and a jump in the shear modulus c₆₆ at the superconducting transition temperature, the available experiments point towards an even-parity chiral superconductor with k_z(k_x ± ik_y)-like Eg symmetry, which has consistently been dismissed based on the quasi-two-dimensional electronic structure of Sr₂RuO₄. Here, we show how the orbital degree of freedom can encode the two-component nature of the E_g order parameter, allowing for a local orbital-antisymmetric spin-triplet state that can be stabilized by on-site Hund's coupling. We find that this exotic E_g state can be energetically stable once a complete, realistic three-dimensional model is considered, within which momentum-dependent spin-orbit coupling terms are key. This state naturally gives rise to Bogoliubov Fermi surfaces.