A decent amount of story; more than something like XCOM, less than something like Wasteland 3. Our story engine draws from a pool of available storylines and selects them based on your power level, character tags on your squad members, and how you've played the game so far, in both story missions and proc-gen missions that work hand-in-hand to feed the story engine and give you opportunities to aid and increase the influence of contacts in your network. When one of our hand-crafted storylines gets selected by the engine, it then weaves in squad members and contacts from your specific playthrough. So every time you play, the story is customized to your playthrough.
We think it's a pretty cool system, and will only continue to become more so as we keep adding new content to the game post-launch.
Thank you very much!