Unfortunately, there is no such option for fanfiction that’s hosted at livejournal or dreamwidth.
However, there is an easy (and free!) method to do so at Instapaper!
First, go get a free account at Instapaper. You will automatically get a bookmark called “read later.” (I installed it on my toolbar and renamed it “saved to instapaper.”)
Now, the main purpose of Instapaper is simply to bookmark stuff you’ve found online that you really want to read, but don’t have the time or inclination once you first come across it. Clicking on the “read later” bookmark will save the article to your instapaper account. When you want to read it, go to your account and click it.
On your account page, to the right, you will see "FOLDERS." You want to create a folder named whatever the piece of fiction is named. (Actually you could name is FRED if you want, but whatever.) Move the piece of fanfiction to that folder. Then, open the folder, so that the only thing displayed in the main screen is that story.
Once you have the piece of fanfiction saved to the folder, go ahead and click the format you want to download it as (“Printable,” “Kindle,” or “ePub”), and blam: it downloads the story, in the chosen format, to your computer. Sync your ereader, and there it is. (Kindle 3 users can also have it emailed directly to their Kindle.) Once you’ve done that, click “archive” or “delete” on the article (piece of fanfiction), and you’re done.
But what if the story is multiple chapters in multiple entries?
No problem—it just takes a little longer.
Make a folder at Instapaper for the story, just like you would for a single-chapter story.
Go to each lj entry of the story, and save each one to your Instapaper account. Make sure you get them all.
Go to your Instapaper account.
There’s a “move” link under each saved entry. “Move” each part of the story to the folder you’ve created. This is the part where you want to be careful: you have to move the last chapter FIRST, then the second-to-last chapter, etc. This saves them the last chapter at the bottom, etc., and the first chapter will be at the top.
Select the folder. You should see all of the chapters listed.
Click on the format you want to download.
If this doesn't work for you (it's possible I've missed out a step just by dint of having done it so often that I haven't simplified it enough), let me know and I can walk you through it.
(N.B.: This probably works at other fanfiction archives, but I haven’t personally tested it, so I can’t say for certain.)