You can't legally store the tweets per twitter's tos. You can store the tweet's id
Supposed to at least
Edit,
I was mistaken. Looks like it's only if you redistribute to third parties,
> The best place to get Twitter Content is directly from Twitter. Consequently, we restrict the redistribution of Twitter Content to third parties. If you provide Twitter Content to third parties, including downloadable datasets or via an API, you may only distribute Tweet IDs, Direct Message IDs, and/or User IDs (except as described below). We also grant special permissions to academic researchers sharing Tweet IDs and User IDs for non-commercial research purposes.
Supposed to at least
Edit,
I was mistaken. Looks like it's only if you redistribute to third parties,
> The best place to get Twitter Content is directly from Twitter. Consequently, we restrict the redistribution of Twitter Content to third parties. If you provide Twitter Content to third parties, including downloadable datasets or via an API, you may only distribute Tweet IDs, Direct Message IDs, and/or User IDs (except as described below). We also grant special permissions to academic researchers sharing Tweet IDs and User IDs for non-commercial research purposes.