Yes, here is how you do it:
1. Placing your Pages and Files into a 'Private' Folder.
2. Setup a Product with 'Grant Access' to the private folder.
4. Make sure you require Login/Registration to Checkout (by placing your checkout pages in a Registration Folder).
Whenever a successful order is placed for the Product by a User, liveSite will allow that User to access the Pages and/or Files in that Folder instantly.