Stack Overflow Enterprise is available for internal use, but it is really meant for larger organizations only.
For personal public sites, Area 51, the Stack Exchange Network staging zone, has a FAQ and it touches upon this:
Can I use Stack Exchange to support my product?
Stack Exchange does not offer a private label service for owning a Q&A site. You are free to create an expert Q&A site in your area of expertise. There is a lot of visibility and credibility to becoming a central figure in an expert community, but Stack Exchange is a product for the Internet community at large, not for corporations or individuals.
Regarding your specific question: it depends. Is your personal website dedicated to a topic that isn't already encompassed by a StackExchange site and would benefit from a Q&A site dedicated to the topic? If so, feel free to propose a site by clicking on the appropriate category on the left site of Area 51 and then 'Propose a new [category] site' near the bottom.
The users of your site would have to register their own StackExchange accounts, however.