It sounds like you may not be aware of the difference between OneDrive and OneDrive for Business. A further distinction can be made by noting that there is the Office 365 version of OneDrive for Business, and the on-premise SharePoint Server version of OneDrive for Business.
Note: In the header or elsewhere on your SharePoint or Office 365 site, ‘OneDrive’ appears as an abbreviation of OneDrive for Business to keep things simpler.
That being clarified, we can now focus in on your question- the answer is yes, if you had a local on-premise SharePoint 2013 or SharePoint 2016 server implementation. You could leverage the OneDrive for Business component of that and keep the documents on your physical, local servers and surface them to users with the standard View/Edit links.
You could further enhance the user experience by incorporating an Office Web Apps Server. Office Web Apps Server is an Office server product that delivers browser-based versions of Word, PowerPoint, Excel, and OneNote. A single Office Web Apps Server farm can support users who access Office files through SharePoint 2013, Lync Server 2013, Exchange Server 2013, shared folders, and websites.