If you wish to download your journal then you can do so using the journal export tool at http://www.livejournal.com/export.bml.
It is currently only possible to export your journal month by month, and only in CSV (comma separated variable) and XML (extensible markup language) formats. While these are useful for keeping an archive, they are not designed to be human-readable. In order to read your entries, you will need to save the file from within your web browser and then use a third party program to read the file. For instance, many spreadsheet and database programs are capable of reading CSV files.
An alternative option is to use a downloadable client, some of which have download or backup features. You can find information on these at http://www.livejournal.com/support/faqbrowse.bml?faqid=158.
Finally, it is possible to download your journal entry by entry, by viewing and saving the web pages. This is the most time consuming option and as such is not recommended, but it is also the simplest. It is also the only method of downloading comments. You can navigate quickly through your comments pages by using the green arrows at the top of the entries, to skip back or forward one entry.
If you wish to print your entries or if you have limited bandwidth, you may wish to set LiveJournal to use the "Lynx" site scheme before you do so, as this offers a simpler, smaller page layout. You may do so at http://www.livejournal.com/setscheme.bml (you may also change back here after you have finished.