I have used FLAG mentioned above and it works very well to pull down whole stories. I use it with my Kindle and you have to sideload the resulting files, but it is easy. It should work just as well for ePUB.
These days I mostly read on my Android phone. I use an app called Fanfiction Reader that pulls down entire stories to read offline. It will supposedly also check for updates to stories you have already downloaded, but I haven't used that feature much.