Is it possible to do something to WP so that I'm able to use the e-mail address as username?
I do not want to download a plugin for this.
You can easily do this, all you have to do is create a user registration page and have the email address field fill both the username and email args to use for the wp_insert_user() function.
WordPress uses the following function when creating users:
Sanitize username stripping out unsafe characters. If $strict is true, only alphanumeric characters plus these: _, space, ., -, *, and @ are returned. Removes tags, octets, entities, and if strict is enabled, will remove all non-ASCII characters. After sanitizing, it passes the username, raw username (the username in the parameter), and the strict parameter as parameters for the filter.
Short answer: Not without a plugin.