Most are. There are six criteria an (unpaid) internship has to meet for it to not be illegal. Essentially, the company offering the internship has to be offering it solely for the benefit of the intern and can derive no real benefit from it themselves. (And a company can't offer an unpaid internship as a way to avoid paying employees for training.)
I think that's the most important point. The six point test is important, but its so vague that it can give people a false sense of safety.
From the linked NYT article:
"'If you’re a for-profit employer or you want to pursue an internship with a for-profit employer, there aren’t going to be many circumstances where you can have an internship and not be paid and still be in compliance with the law,' said Nancy J. Leppink, the acting director of the department’s wage and hour division. "
"and can derive no real benefit from it themselves"
I would think that would be pretty impossible to conform to no matter how you structured the intern's day/tasks. Unless you simply provided them a desk in which to study, or worked and let them observe with no interaction I would think sticking to this rule would be difficult.
The full list (linked from the article) is here: http://www.theeagleonline.com/news/story/new-study-raises-qu...