Ugh... You're in a pickle! (As "they" say in certain places.) :)
The most obvious thing that would normally need to be done is to upgrade to the latest release. With this being a version 3.xx instance, upgrading would be just as time-consuming as rebuilding the website. Unfortunately, no matter the software, if it just sits there without regular maintenance (e.g., upgrades and updates), it will indeed break over time. Seemingly (and sadly), it will indeed come out of nowhere as well.
What's most likely in your case is that a Windows Update is causing the code to break. This is not something that would allow you to apply a quick fix.
If you have the expertise to rebuild the website in a current release of DNN, this would be your best and cheapest option to fix this now, as well and protect your website investment over the long term. This is because much of the DNN work is already done. It's a matter of finding feature parity with today's features and extension options.
If you don't have the resources to do this yourself, you can find an agency to help you as well. :)