Magnetization and neutron diffraction measurements on polycrystalline BaCuO2+x revealed a combination of magnetic behaviors. The Cu6 ring clusters and Cu18 sphere clusters in this compound had ferromagnetic ground states with large spins 3 and 9, respectively. The Cu6 rings ordered antiferromagnetically below the Neel temperature T(N) = 15 +/- 0.5 kelvin, whereas the Cu18 spheres remained paramagnetic down to 2 kelvin. The ordered moment below T(N) was 0.89(5) Bohr magnetons per Cu in the Cu6 rings, demonstrating that quantum fluctuation effects are small in these atomic clusters. The Cu18 clusters are predicted to exhibit ferromagnetic intercluster order below about 1 kelvin.