When the answer to that question no longer matters to you, or when people are unable to get to you by using that word, you'll be an adult. When I was 16, I acted very grownup so I would be treated like an adult. It mattered to me what they thought of me. When I was 18, I made a point of saying I was an adult and demanded to be treated as such. Somewhere between then and now, when I hang around kids all day (I'm a teacher), dress wearing novelty shirts every day, and I generally act like a teenager, I became an adult. Today, I get a lot of "Act your age", and "You should be an adult by now". It amuses me that these people, who grew up believing that being an adult meant doing whatever one wanted, now are convinced that being an adult means doing whatever one is expected to.
When you find the place where the only opinion of you that matters is yours, you'll be an adult. Whether that happens at 18 or 81.