I think antioxidants are quite important. You can't eliminate all sources of free radicals in the body, nor should you want to, so allowing natural mitigation from antioxidants seems like it would be fairly important. I've always tended to shop around the periphery of the grocery store (i.e., primarily fresh produce, meat, and maybe a little dairy.) Eating a diet of less processed foods naturally seems to provide reasonable sources of antioxidants, which at least makes some intuitive sense as our bodies evolved for millennia to thrive on these food sources.