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I'll try to spell it out for you.
The article is not saying that classification methods don't work, and isn't disparaging any particular method or combination of methods. It is not even stating that you cannot achieve arbitrarily small error. The article is ONLY saying that you cannot reliably estimate the resulting classifier accuracy (out in the real world) using cross validation or bootstrapping on your dataset.
If your data sanple is large and diverse enough, you could. But you just won't ever know if that is the case or not.
Personally, I think their 'main conclusion' stretches the point a bit, probably for sensationalism's sake.