This could be done with anything you are comfortable with. I would personally use Ruby on Rails for this task. As this is what I know the best. So here what you can use in Rails to do that:

- To manage your accounts you can use different auth gems: https://www.ruby-toolbox.com/categories/rails_authentication here is the big list of them.

- To make a billing system you can use: https://stripe.com/

- There is a really easy approach for making REST API in Rails