I have Microsoft entourage on my MacBook Air, with thousands of e-mails on it. Without thinking I upgraded to the new Catalina OS and now I can't get in to Entourage. Is there any way of accessing all of my e-mails?

Entourage was last used in Office 2008. It changed to Outlook in Office 2011. The only way to get at the contents of an Entourage database is with Microsoft's email client.

If you made a full backup before upgrading to Catalina, you can restore it to return to that point. Otherwise, you have to do two steps to move forward.

1. You need Office 2011 to import the Entourage database. Newer versions of Office do not offer a way to import email data from Office 2008 or older. If you know anyone using Office 2011, have them import your 2008 database to a new user account (so it doesn't get mixed with theirs).
2. The updated import will have the same name as in 2008. It will the be Main Identity folder in the active user account. Bring that imported/converted 2011 database back to your Mac.
3. Purchase either the one-time cost, perpetual license Office 2019, or the subscription version Office 365. Make sure to purchase an option that includes Outlook.
4. Import the Office 2011 data in Outlook.

After that, the Main Identity folder is no longer used. Everything will be in this folder:

/Users/your_account/Library/Group Containers/UBF8T346G9.Office

Generally, the only thing that doesn't come over are the settings you use to connect to your email account. The password will have to be entered and saved to your keychain.

You must have an old 32bit version of Office, probably Home and Student 2011? If so you cant get into Word or Excel either.

Primarily Microsoft forum is place to go.

Are the email accounts stored online? Can you not add the accounts to another mail client like Mail? Or log into them using Safari?

otherwise you may have to subscribe Office 365 and migrate to Outlook app (plenty of info on Google)

See Kurt Lang's reply