We don't need no plants antioxidants!

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- probably! As hinted in the following study:

Dietary (Poly)phenolics in Human Health: Structures, Bioavailability, and Evidence of Protective Effects Against Chronic Diseases", by
Daniele Del Rio, et al., Antioxid Redox Signal. 2013 May 10; 18(14): 1818–1892.

Very poor absorption of plants' poly-phenolic anti-oxidants! From milli-Mole in food down to nano-Mole concentration in the body! Quickly destroyed and eliminated by the body! After decades of hypothesizing and speculations - no direct evidence of health benefits!

Quotes:

...exist in planta, at concentrations in the low-μM-to-mM range. However, after ingestion, dietary (poly)phenolics appear in the circulatory system not as the parent compounds, but as phase II metabolites, and their presence in plasma after dietary intake rarely exceeds nM concentrations. Substantial quantities of both the parent compounds and their metabolites pass to the colon where they are degraded by the action of the local microbiota,
...
Evidence relating to the anticancer effects of (poly)phenols is limited. The majority of available clinical evidence has been with green tea/(poly)phenols in populations at a high risk of cancer development, with results proving inconclusive.
...
While a few of studies suggest that (poly)phenol-rich foods prevent lymphocyte DNA damage, no direct link with a decrease in cancer risk can be established from those studies.
...
As yet, it is too premature to consider the potential use of (poly)phenolic compounds as therapeutic agents.

Updated 22/12/2018

This is probably the main reason why dietary anti-oxidants don't work:

"Antioxidants prevent health-promoting effects of physical exercise in humans"
by Michael Ristow, et. al., PNAS May 26, 2009 106 (21) 8665-8670; March 31, 2009

...We evaluated the effects of a combination of vitamin C (1000 mg/day) and vitamin E (400 IU/day) on insulin sensitivity as measured by glucose infusion rates (GIR) during a hyperinsulinemic, euglycemic clamp in previously untrained (n = 19) and pretrained (n = 20) healthy young men. Before and after a 4 week intervention of physical exercise, GIR was determined, and muscle biopsies for gene expression analyses as well as plasma samples were obtained to compare changes over baseline and potential influences of vitamins on exercise effects. Exercise increased parameters of insulin sensitivity (GIR and plasma adiponectin) only in the absence of antioxidants...
...
Consistent with the concept of mitohormesis, exercise-induced oxidative stress ameliorates insulin resistance and causes an adaptive response promoting endogenous antioxidant defense capacity. Supplementation with antioxidants may preclude these health-promoting effects of exercise in humans.

Anti-oxidants may promote rather than fight cancer!

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Hold off those berries and vitamins, suggests DNA discoverer James Watson, in today's article titled "DNA pioneer James Watson takes aim at 'cancer establishments'" article.

From wiki

Quotes:

On the $100 million U.S. project to determine the DNA changes that drive nine forms of cancer: It is "not likely to produce the truly breakthrough drugs that we now so desperately need," Watson argued. On the idea that antioxidants such as those in colorful berries fight cancer: "The time has come to seriously ask whether antioxidant use much more likely causes than prevents cancer."

...

That is why Watson advocates a different approach: targeting features that all cancer cells, especially those in metastatic cancers, have in common.

One such commonality is oxygen radicals. Those forms of oxygen rip apart other components of cells, such as DNA. That is why antioxidants, which have become near-ubiquitous additives in grocery foods from snack bars to soda, are thought to be healthful: they mop up damaging oxygen radicals.

That simple picture becomes more complicated, however, once cancer is present. Radiation therapy and many chemotherapies kill cancer cells by generating oxygen radicals, which trigger cell suicide. If a cancer patient is binging on berries and other antioxidants, it can actually keep therapies from working, Watson proposed.

"Everyone thought antioxidants were great," he said. "But I'm saying they can prevent us from killing cancer cells."