From Twitter TOS :
Sending, publishing or displaying Content on or through the Services, you provide us with a worldwide non-exclusive license (with the right to sublicense) to use, copy, reproduce, process, adapt, modify, publish, transmit, display and distribute such content in all ways media or dissemination methods (now known or developed later).
[TIP] This license you authorize us to make your tweets available to the rest of the world and let others do the same. But what yours belongs to you - you own your content.
In other words, yes, you can store them and reproduce them as you wish. This page also refers to API rules that have several requirements, for example, attribute tweets, do not edit them to change their meaning, and obtain permission before using them on a commercial basis (for example, on T-shirts).