The problem here is that the things that I'm writing in popup.js with window.webkitRequestFileSystem can be read from there, but I can not be read from content.js.
PS: the code is the same in both files
manifest.json
 {
  "manifest_version": 2,
  "browser_action": {
    "default_popup": "action.html"
  },
  "content_scripts": [
    {
      "js": ["content.js"]
    }
  ],
  "background": {
    "scripts": ["background.js"]
  },
  "permissions": [
    "unlimitedStorage",
  ]
}
action.html
// Here is the popup.js file included
popup.js
window.webkitRequestFileSystem(window.PERSISTENT, 0, readFromFileStorage, errorHandler);
function readFromFileStorage(fs) {
  fs.root.getFile('file.txt', {}, function(fileEntry) {
    fileEntry.file(function(file) {
       var reader = new FileReader();
        reader.onloadend = function(e) {
          console.log(this.result);
        };
       reader.readAsText(file);
    }, errorHandler);
  }, errorHandler);
}
content.js
window.webkitRequestFileSystem(window.PERSISTENT, 0, readFromFileStorage, errorHandler);
// the function 'readFromFileStorage' is the same as in "popup.js"
 
    