In small doses of exercise, I wouldn't sweat the need for extra antioxidants outside what is found in food. The body can make it's own potent antioxidants also such as glutathione.
In the end....a little stress makes the body smarter and stronger...that whole hormesis thing......too much is not good of course.
Anything taken to excess that may down-regulate internal processes (and never provide the small stressors the body needs to respond to) may not be a good thing. Kind of like always taking a test with the answer sheet in front of you....won't really make you that much smarter.