Would you consider a DMS "off the shelf"?
I'll admit it isn't cheap, but we use Filesite / Worksite / Autonomy (or whatever name its being called this month!) and it is completely integrated into Outlook / Exchange / Word / etc.
An email comes in, with attachments, and I can file the email, and its attachments, direct into Filesite from Outlook. It attaches against a Client, or a "Matter" number. We have a separate 3rd party App for all our Matters (legal cases, but the could be anything associate with Clients / Customers - Invoices, CRM, or something else). In Outlook (under Public Folders) we have a complete tree structure for all our clients, and under each client a folder for each Matter, and then for each Matter we have some folders for Accounts / Projects / etc. [bespoke to us / the way we work]. So I can easily "assign" the Email to the appropriate Client / Matter / Sub-Folder. i.e. Autonomy and our 3rd party Records Management software is completely integrated, and "transparent" to the users
Most of the time the system will "guess" the correct folder, or at least give me a "shortlist" to choose from.
Same if I do "Save As" from Word - it will save, by default, into the Document Management System rather than to my C: drive! and predict the appropriate folder.
The only paper we have in the office is some HR stuff that we are obliged, by law, to hold. Everything else, coming in the post, is scanned and shredded - but these days the majority of incoming documents are by EMail, not post, so year-by-year we downgrade the size of our scanners they used to be the size of a small room ...