Yes, this is possible with Business Catalyst.
You can publish the site and use secure zone, include the pages in secure zone so that accessing the pages would requires users to login.
These documents would help you :
Additional information :
Thanks, that covers the basics, just 2 clarifications since I jumped through the bcguru videos.
It said you can make pages with user specific data (where they can see orders, form submissions etc.) but is there a way where I could make specific pages. So if John A. logged in he would go to a landing page displaying data or files specifically for him and if Jane B. logged in she would get a different landing page with specific data and files for her?
Also in that tutorial it talked about setting up accounts/paid accounts but with the regular account is there a way to set it so for one to become active they would need an admin to authorize it?
Yes, you can setup both.
These docuemnts defines the steps with which you can achieve this :
Web app that requires payment :