There might be other or better solutions as moshu listed, but surely we would not say something like this is 'impossible' with WordPress. There are many ways.
One idea. Each edition could be a WordPress "Page." When the day is over you can change the Page to "draft" instead of "Publish" and it will disappear from your site. You would either create the page manually each day (hey, it's easier than laying-out a real newspaper and printing it on a news press).
Another way. If you are happy to let yesterday's and last month's editions remain on the site in older Archives. Then you can basically use WordPress as-is, you post all the daily articles, and you use htaccess or a plugin to make your Monthly or Weekly or Daily Archive (automatically created by WP as-is) as your homepage. Therefore the homepage shows always the posts of today only.
Another way. Make a custom-query for your homepage which calls only posts with a date that matches today's date. Do not show links to older archives if you don't want them.
Another way. Use WordPress as-is and get the Post Control plugin, use it to manually publish and unpublish multiple posts in 1 admin page.
Just look at the new site of NewYorkTimes.com if you need more ideas. I don't know if they use WP now but their site could be made with WP.