The above answer is correct: no they can't function independently. I want to build on that answer with a couple additional points.
Consider that your heart beats while you hold your breath - but it needs oxygen, so if your lungs stop for a long time they aren't exchanging O2 and CO2 so your heart muscle will eventually stop. Conversely if your heart stops, you're on your way to dying and your neurological breathing mechanisms will eventually stop, as will the muscles that move air in/out.
So again as the above answer discusses, fundamentally, if one stops the other will soon stop too. Then soon thereafter the rest of the body too, i.e. death. Neither organ could last very long on its own without mechanical support to get it going again.
For the question
can the lungs take in oxygen, and, not having anywhere to put it, breathe it back out?
the answer is that if blood stops circulating, your lungs will soon stop moving air in/out - but even then there IS equalization of O2 and CO2 partial pressures between the blood and the little bit of air left in the lungs, even if neither is moving. But that doesn't do any good for the person, who is basically dying, unless the blood starts circulating again - by CPR or a machine or on its own.
CPR circulates blood primarily, but also pressing on the chest moves a little bit of air in/out of the lungs. That is why the algorithm for CPR removed rescue breathing for sudden collapse / cardiac arrest in teens & adults outside of the healthcare setting. "Hands only CPR." Compressions are precious in circulating blood to the brain, and seconds of them are lost when doing breathing. EMTs and Paramedics outside the hospital are a different story, as they are skilled medical professionals, and might use a bag device to help breathing.
It is that principle of equalization of O2 and CO2 between blood and air that makes the tiny bit of air that goes in/out of the lungs during chest compressions good enough to not lose precious compressions.