There is already a fine answer - this is just a minor addition. Some people might prefer Markdown for the posts rather than HTML. The other query uses Body from the Posts table - this contains HTML. Markdown can be obtained using the PostHistory table.
- Here is a query which returns Markdown for a single post.
- Here you can get Markdown for posts of a specific user: only questions, only answers and both.
- A similar query for posts from a deleted user: only questions, only answers and both.
I will repeat a few of the usual caveats:
- SEDE is only updated once a week - so you won't get very recent posts.
- You don't get the deleted posts in this way.
- The results of a query are limited to 50000 rows - but so far even the most prolific posters on MO are below 50k posts (or 50k comments) - so this won't be problem for MO in the near future.
Naturally, it is possible to do much more - if you have a look at rene's answer to Dump of my own Stack Exchange content, you can see that there is a query which collects contributions over all sites, creates a HTML file consisting of all results. And also an alternative query, which returns Markdown - again for posts from all sites in the network.