I have a chrome extension (not App),Its a mashup of data from free third parties. I don't maintain a server for my extension.
Can i use Google drive to save user's data using users account, so it can be shared by different computers?
Yes, you can! Check out this other StackOverflow answer that explains how to use the google-api-javascript-client library to send data to the Google Drive API.
You might also be interested in this blog post explaining how to mash up Web Intents and the Google Drive API.