You don't need to pay for Google storage to use Google Mail. I don't and Gmail works just fine. However, if your free storage is full and you don't clear some of it or else pay for more, there won't be any space for new mail and your Gmail will not work.
See the following from my previous post. You can use that information to see what you are using the Google storage for and to possibly clear some of I:
The following article from Google has information on learning what you have in their cloud storage and information on how you may be able to clear some of that space. If you are in the U.S., you get a minimum of 15GB of Google cloud storage free> https://support.google.com/drive/answer/6374270?hl=en