Vitamin C has been very widely studied and if it really was the wonder drug some people make it out be then I suspect we'd know about it by now. It does appear to be worth taking once you have a cold as it can lessen the duration in some people.
The evidence base on Vit C preventing colds goes beyond lack of evidence. Test after test and the subsequent reviews show that it does not prevent colds. If it did, it would have shown up on these tests. I think we can safely say it doesn't prevent colds nor flu.
The situation with antioxidants is far more complicated. The reasons to believe that taking lots of antioxidants daily is good for you are not strongly backed up with evidence but mostly based on snippets of research and relatively isolated application of physiology. Lots of things sound good in theory but don't work in practice. In the body this is even more the case because we cannot model the body properly, not by a long way. So we cannot reliably predict the effects of chemicals based on our knowledge of biology. Nor are in vitro or animal studies reliable predictors. In fact even human trials are not terribly reliable either but obviously are the best we can do. So without proper trials we just don't know what effects these things really have especially in the long term. This is why decent doctors and medical scientists do not recommend things without sufficient evidence and change their views when that evidence changes. That is why medical representative groups like BMA or AMA don't recommend widespread daily supplementation. Alt heath practitioners don't tend to take such a responsible attitude and jump on isolated papers to further their cause. This article is illuminating:
http://www.sciencebasedmedicine.org/index.php/antioxidant-hype-and-reality/
Lack of evidence doesn't necessarily prove something has no value but it does prove we don't know that it does and it does indicate that it might even be harmful.
Tumeric is interesting. It does have anti-inflammatory properties. I took it for a while for my back problems. It didn't noticeably help as it happens but I thought it worth a try given the problems I had. Whether it contributes to low cancer or Alzheimers is just speculation. People also credit this to green tea, low meat diets, lack of dairy products and a host of other dietary and environmental differences. Most of the studies showing possible positive effects end with the line "worthy of further investigation" I.e. they are preliminary and the majority of preliminary research is later shown to be wrong (
http://theness.com/neurologicablog/index.php/are-most-medical-studies-wrong/). It looks promising but we don't know the full story yet. I think the thing to remember is that if it does work then it's a drug. And all drugs have side effects and so should only be taken to treat a specific condition. We don't know all the side effects of curcumin yet but we do know some of them. A quote here from McCormick Science Institute:
"Curcumin may accelerate reactive oxygen species formation and
damage lung tissue of smokers and ex-smokers.
Other toxic and carcinogenic properties of curcumin have been described.(31) The mechanisms underlying curcumin’s anticancer actions include inhibiting carcinogen metabolism; suppressing tumor invasion and metastasis; disrupting regulation of transcription factors, growth factors, protein kinases, or mTOR; and modulating levels of inflammatory cytokines that control cell proliferation, angiogenesis, and apoptosis.(4,5,29,30,40-43) In this regard, 2 reports of in vitro studies indicate that curcumin may induce degradation of the tumor suppressor protein p53 and inhibit its activity,(44,45)
results that raise concerns about safety"
These negatives are also preliminary and may turn out to be nothing, just like the positives. If you look at the McCormick overview you'll see that they put P for Preliminary against every section. I.e. we don't know for sure right now. Hopefully at some point we will and it will be good news. Until then I wouldn't recommend taking large amounts on a daily basis just in case that tumor suppressing protein turns out to be important.
Of course individuals are free to attempt to weigh up evidence with regards to their own welfare and do what they like.
The Neurologica blog I linked to above is superb. It's written by the Director of General Neurology at Yale University School of Medicine who also has an interest in rational thinking. Here is another article which talks about complexity of biological systems and is quite pertinent to the above:
http://theness.com/neurologicablog/index.php/calcium-and-the-law-of-unintended-consequences/ - something as apparently harmless as calcium supplementation has some quite shocking downsides.