Hey @helenrichards, welcome to the Dropbox Community and happy Monday!
It's true that there isn't a direct way to backup your emails to Dropbox, but if you find a way to download them and convert them locally on your device to a file type that's supported, you might be able to copy or move them in Dropbox.
You might also want to check out Apple's forums for possible ways to backup your emails.
I know it's not much, but I hope this helps!