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It can certainly be done, but the solution details are probably beyond the scope of this forum.

There are several ways to have custom content in widgets. One way is to use My Custom Widgets plugin.

Using that, you could code a custom query to select all posts having a date less than the date of the current post, and list the links to them.

You did not say what you wanted to do if you were on a page that showed multiple posts, but you should consider that when designing the widget.