We propose a minimal model resolving a puzzle of enigmatic correlations observed in sodium-rich NaxCoO2 where one expects a simple, free motion of the dilute S = 1 / 2 holes doped into a band insulator NaCoO2. The model also predicts singlet superconductivity at experimentally observed compositions. The model is based on a key property of cobalt oxides - the spin-state quasidegeneracy of CoO6 octahedral complex -leading to an unusual physics of, e.g., LaCoO3. We show that correlated hopping between t2g and eg states leads to the spin-polaron physics at x~1, and to an extended s-wave pairing at larger doping when coherent fermionic bands are formed.