We have used the Submillimeter Wave Astronomy Satellite (SWAS) to carry out deep integrations on the NJ = 33 → 12 transition of O2 in a variety of Galactic molecular clouds. We here report no convincing detection in an initial set of observations of 20 sources. We compare O2 integrated intensities with those of C18O in a similarly sized beam and obtain 3 σ upper limits for the O2/C18O abundance ratio ≤ 2.3 in four clouds and ≤ 3.6 in five additional clouds. Our lowest individual limit corresponds to N(O2)/N(H2) < 2.6 × 10-7 (3 σ). A combination of data from nine sources yields N(O2)/N(H2) = [0.33 ± 1.6 (3 σ)] × 10-7. These low limits, characterizing a variety of clouds in different environments at different Galactocentric radii, indicate that O2 is not a major constituent of molecular clouds and is not an important coolant. The abundance of O2 is significantly lower than predicted by steady state single-component chemical models. The present results are best understood in the context of cloud chemical and dynamical models that include the interaction of gas-phase molecules and grain surfaces and/or circulation of material between well-shielded and essentially unshielded regions. This circulation may be powered by turbulence or other driving forces that effectively keep molecular clouds chemically unevolved.