Twitter has a simple process to download tweets and user data.
Here is how to do it :
Go to Settings from the gear icon.
Scroll down and click on “Request your archive”.
A confirmation pop-up will appear, click “Close”:
At the email provided when signing up for Twitter, the download link will be available, click “Go now” to start downloading the Twitter archives:
The downloaded data will be labelled as “tweets.zip”. Extract it, the archives will be something as shown below
To access these archives locally, simply open the index.html file in browser, all the earlier tweets will be available for viewing offline on a year wise basis.
Store this archive folder somewhere safe as a backup.
By the way, this entire process is very similar to how Facebook data can be backed up. Here is an earlier article related to that.