However, what if I do want an access token, either to get around the new paging restrictions for anonymous users, to read personal information (e.g. my inbox) or to get write access? Do I have to set up a website so I can use the OAuth flow as described in the documentation?
You can use any domain (e.g. stackapps.com) as OAuth domain; you don't need to control it.
Open the following URL in your browser (add the actual client id from the application page, and the scope with what you actually need - for details, see the documentation): https://stackoverflow.com/oauth?client_id=...&scope=read_inbox,no_expiry,write_access,private_info&redirect_uri=https://stackapps.com
After authenticating, you'll get redirected to https://stackapps.com/?code=...; the code ends in )).
Execute the following command in a terminal (add the client id and client secret from the application page and the code from the previous step): curl -d "client_id=...&client_secret=...&code=...&redirect_uri=https://stackapps.com" -X POST https://stackoverflow.com/oauth/access_token
The result is a simple access_token=... and the token also ends in )).