The bottom of every page has a link to the blog which gives some more details regarding attribution considering the cc-by-sa license:
All the content [is] ... intended to be shared and remixed.
If you republish this content, we require that you:
- Visually indicate that the content is from Stack Overflow or the Stack Exchange network in some way. It doesn’t have to be obnoxious; a discreet text blurb is fine.
- Hyperlink directly to the original question on the source site (e.g., http://stackoverflow.com/questions/12345)
- Show the author names for every question and answer
- Hyperlink each author name directly back to their user profile page on the source site (e.g., http://stackoverflow.com/users/12345/username)
So, yes, you can presumably build a website or app from this, as long as you provide attribution as specified above (and use the same license).
Although, if you try to make it look like an "official" app, I expect you might have some problems (but that probably has more to do with usage of the name and logos and such).
If you're storing it either just for yourself or also with proper attribution if you end up using it, it's presumably also fine (they do provide data dumps, so it's not like they're trying to prevent this).