conundrum: oxygenation vs. oxidation

i know just enough science to be dangerous, and i do a lot of reading.  

(i would say i'm a generalist, in that i know a little bit about a lot of things, which makes for an interesting life.  and part of that curiosity is when to seek out a person who specializes in an area.)

i got to thinking about the nature of oxygen recently, due to part of a book i was reading.  and i realised yet another thing i don't know.

namely this:  we can't survive without oxygen, and yet oxidation destroys.

the irony!  

i mean, leave a jar of oil or wine open and it will start to go bad.  an apple once peeled goes brown.

how can it be that the very thing that keeps us alive is also breaking down life forms all around us?

we breathe and exercise to oxygenate our cells, right.  

so why doesn't that also destroy them?  

(clearly i need to go back to high school chemistry.)

any insights other than it being high time to power up on antioxidants?