Here are several things you can do to back up your LJ posts, their accompanying comments, and other data.
1) Use
LiveJournal's export tool.
This will allow you to download the text of your entries in a format that can be reuploaded later. Images and comments are not included.
2) If you have a Dreamwidth account, use the
Dreamwidth importer.
This works for some people and not for others. If it works, it can import your journal entries and their accompanying comments to Dreamwidth, along with your tags, profile, and icons. [ETA: See
byslantedlight's notes on images below. Further ETA: As of 11 March Dreamwidth imports are working for some people for whom they failed before.]
3) Use
BlogBooker.
This site can generate a PDF or DOC that includes all of your journal entries with their accompanying comments. You can make three documents for free, each containing entries from a period of either 1 year (for PDF) or 6 months (for DOC). If this is not enough to back up your whole journal, there are paid options that you can consider. (Note that using BlogBooker requires that you share your LJ password. You might want to change it after downloading your back-ups.) [ETA: Some people are reporting that BlogBooker is having issues due to high traffic.]
NEW!
4) If you have a blog that uses Wordpress, take advantage of Wordpress's LiveJournal import tool.
Wordpress blogs have a built-in tool for importing from LiveJournal. Instructions for using it can be found in
this Wordpress support article. This method is supposed to import both entries and comments, but I had trouble with the latter. Images are not imported, but rather embedded in the imported entries using links to LiveJournal. If you don't have a blog that uses Wordpress, you can start one easily by signing up at
Wordpress.com. (Thank you to
mikesmaddie for sharing this method.)
I hope this is useful to some people. If you know of other good methods, please share them in a comment.