If you are logged in to your Blogger account, you can easily download a dump of your website by clicking on the Settings → Other → Export Blog link. (Source.)
How can I do this automatically each day on Linux, from a cron job?
First you need to do a login with curl and then you can simply invoke another call to the download link.
I would follow this article to accomplish this. The trick is to find out which URL you need to call to export your blog.
What I did is:
You will see the download happening, and you will see a new record in the network tab. Mine is : http://draft.blogger.com/feeds/7135654868651822450/archive (if not authenticated you will get 404).
(update by Mark Harrison) Based on the information above, here is a script that will download the xml dump of the blog. (end update)
FEED=12312312312312313 GMAILemail@example.com echo -n google passwd: stty -echo read GPASS stty echo AUTH=`curl 2>/dev/null https://www.google.com/accounts/ClientLogin \ -d Email=$GMAIL \ -d Passwd=$GPASS \ -d accountType=GOOGLE \ -d source=blogix \ -d service=blogger \ | grep '^Auth=' | cut -c 6-` curl >blog.xml \ -H "Authorization: GoogleLogin auth=$AUTH" \ http://www.blogger.com/feeds/$FEED/archive
As someone who uses Backupify, IFTTT, Zapier and a handful of other cloud connecting sites, I think in your case, a one time dump plus a IFTTT -> Dropbox for posts and files would be your best bet. To verify it's working, you could also have it send you an email with your post.
IFTTT has been around for years now and has always been reliable. They are planning on charging at some point, but I can't imagine you would fall under that for just this use case alone.
Backupify will charge you right off the bat.