Rae Hoffman wrote a post at Search Engine Land about the opportunities you have if one of the links you have points to a 404. You can create a relevant 301 redirect, a user-friendly 404 page, or a custom non-404 error page that catches broken links. Rae goes through the various pros and cons of each strategy in a very useful and informative post.
It's really important for you to reclaim those broken links. Vanessa Fox comments on the Sphinn discussion that you can check your 404s using Google Webmaster Tools. Internal links can be fixed, but you should contact the people behind the external links if you can.
An important note is to avoid getting those 404 pages indexed as Vanessa and Rae agree:
One problem with redirecting all requests that would ordinarily return a 404 response with a 301 to a 200 is that search engine bots will think all non-existent pages are real pages and lots of badness can happen because of that. As Rae notes, your 404 pages will get indexed.
Therefore, you may want to be careful when implementing these strategies.
Forum discussion continues at Sphinn.