What is the legal age to run a site?

[ZiNgA BuRgA]

Laws differ depending on where you live.  In Australia, under 18's need parents' permission to use the internet (why all schools require parents sign a form before the student can get internet access).
Apart from that, I don't believe there's a minimum age to operate a website (even if there is, I don't see it being enforced, so it's useless anyway).

For credit cards, one probably has to be at an age where one can accept contracts (usually 18+) or have the parent accept them.

If you're running an 18+ site, the general methodology is to provide a warning before entering the website.