Yes, you can create any number of website within a single liveSite, each with their own menus, page layouts, and design.
Keep in mind, there are some limitations to this approach:
1. System Pages such as the "My Account" page are shared across a liveSite.
2. The same Site Admins, Designers, and Managers are shared across a liveSite.
3. If you are collecting payments online, then the same domain name (because of SSL Certificate), and the same depository bank account are shared across a liveSite.