Is it possible to crawl private pages that require a login to access?

It depends on the login mechanism, but, in most cases, yes. You can find more information on what’s possible here: https://www.algolia.com/doc/tools/crawler/apis/configuration/login/

The crawler supports multiple ways to authenticate for protected websites:

• With a direct request with the credentials to your login endpoint, like a standard curl command (fetchRequest).
  By emulating a web browser, loading your login page, entering the credentials and validating the login form (browserRequest).
• OAuth 2.0