The government sets up a website where you can upload the work you want to copyright and register yourself as the owner. If it's something like software, you must upload the code. It wouldn't be visible to anyone, but would be there in case the copyright expires.
If someone wants to use your work, they can contact you though the website.
If you don't reply (edit: this should say acknowledge) in some reasonable amount of time (say three months), then the copyright is considered abandoned and the work enters the public domain.
For existing works, give people say two years to upload and register. Anything not registered (or at least marked as "in dispute") becomes public domain.
This at least solves the abandoned works problem, as well as the archiving problem. After 100ish years, we'd have a copy of every work as it enters the public domain.