Where does Chrome's Gmail Offline store its files on my Mac?
Gmail Offline usually stores its files under your profile path.  

You can find your profile path by typing chrome://version in the address bar, and searching for Profile Path.  

The path is usually: 


  /Users/%username%/Library/Application Support/Google/Chrome/Default/  


Alternatively, if you are logged in under the account you want to explore, the path is:


  ~/Library/Application Support/Google/Chrome/Default/


If you have a Chrome user profile other than the default one, replace Default with that profile name.  

You should then look at the "Local Storage" and "Local Extension Settings" directories under that path.