Harry's new restaurant has really _____ according to the woman.

The phrase "taken off" in this context means that Harry's restaurant has become successful or has gained popularity. This expression is commonly used to indicate that a business or project is starting to thrive, attracting customers, and overall experiencing a positive upward trend. The woman's comment suggests that there is recognition or acknowledgment of the restaurant's achievement.

In contrast, other options don't convey this sense of success. "Failed" would imply that the restaurant is not doing well, and "closed" would indicate that it is no longer in operation, both of which contradict the positive implication of the woman's statement. "Begun" is too vague and does not effectively express the notion of success or growth that "taken off" does. Thus, "taken off" is the most appropriate choice in this scenario, highlighting the restaurant's successful trajectory.