Oh yay I can finally help!
Yes, this is very much possible and actually easy. We had to do this exact thing, we wanted to open as-built plans as PDF documents with a file:// link. IE (rather questionably and unfortunately in terms of security) will open a file:// link. Chrome and Firefox will not. So, you create a Virtual Directory on your server, such as with Microsoft's IIS. Then, you link that Virtual Directory to the network path, the file path. So it would be mysite.site.local/TheFolder/file1.pdf where TheFolder is the virtual directory that really links to \\fileserver.yourdomain.local\TheFolder\file1.pdf
For more see.. Why doesn't a UNC Link in Pop Up display image? etc..
Now.. another handy trick (and yes this works with the approach above) is in ArcGIS Online web maps you can put a link or multiple file links in, as a "image". I screenshotted (with the Snip) the word "Property Card" to create links to our property cards. The 'word' Property Card, as a PNG image, appears at the bottom of the web map's pop now. You can make multiple "links" with this workflow. It works great! Our users LOVE it! Oh I should note, it goes to the right property card below because I put {PIN} at the end of the URL and that is how it looks it up on the backend.