There are several ways of doing this, including saving your posts as a *.doc/Word file, or turning your blog into a book. (HT: Mary at The Vocation of Motherhood)
Your blogging host also should have instructions in their FAQ or Help file that will enable you to do this (almost) painlessly. Some of the most popular ones:
- Basic Typepad instructions for making backups, from Neville Hobson.
- If you've got a blog at Wordpress.com, follow Lorelle's instructions.
- For the Blogger blogger, here's how.
- For wordpress (not wordpress.com) users, you need to go into cPanel/
phpMyadmin and backup the database from there. Detailed instructions can be found here.