Here is the way i used chrome.storage in my packaged chrome application. It supports large data
- I Added storage permissions in my manifest.json
- I used chrome.storage.local.set in my js to store data in my browser local storage
manifest.json
"permissions": [
        "notifications",
        "storage"
    ],
script.js- to set the data to local storage
 chrome.storage.local.set({
                          'loginToken': response.data.loginToken,
                           'role':response.data.role
                      }, function() {
                       // console.log("The value stored was: " + loginToken);
                   });
script.js - to get the data from local storage
chrome.storage.local.get('loginToken', function(result) {
                                if (!(result.loginToken)) {
                                    alert("please login with your details!!");
                                }
                            })
if we build hosted chrome app. we can use window.localStorage
please refer to this document for the further details.
https://developer.chrome.com/extensions/storage