Since I put quite a bit of effort into my blog and treat it almost like a book-in-progress -- and since self-publishing makes me productive --- I print out my blog posts, three-hole-punch them, and keep them in a binder. That way I've also got something to show the grandkids :D someday, when perhaps the internet has changed beyond recognition and they want to see what it looked like way back when.
No, I don't print them out. My husband is very anti-paper. But he does back up our server and whatever else needs to be backed up.
I find that it is even fun to go back myself and see what I have written in years past. I did print out the blog posts from our honeymoon in Italy five years ago and I scrapbooked them in my honeymoon album.
I don't print out my posts, but I backup my blogs just in case. I also save copies of my posts to another portable drive - another just in case. Now, I do have some posts I have printed - like when my daughter had her "stern" talk with me and when my son melted the mac n cheese in the microwave - the funny posts that I want my kids to read when they are older. Those are keepers!