Depending on which way you want to do the networking, I did have a solution a few years ago involving VPC 2007, Windows 95, Interlnk and a parallel cable.

The old machine was connected to the modern machine with a parallel cable. On the old machine, I ran INTERSVR, then on the modern machine I installed Windows 95 inside a VPC 2007 VM with parallel port passthrough enabled. Within Windows 95 I enabled the option to let it use DOS mode storage drivers. This lets you install INTERLNK on Windows 95 and see the old machine's hard drive from within it. This INTERLNK connection in turn could be shared on the Windows 95 VM and then it could be accessed by the host machine as a network share (or indeed from any other machine on the network).

End result was being able to read/write to the old hard drive directly from a modern OS with the virtual 95 machine acting as a bridge.

Whilst this was fine for backing up old hard disks or moving downloaded files over to the old machine, it is unfortunately the 'wrong way around' if you were hoping to browse the modern machine's resources from the old machine, and of course INTERSVR ties up the old machine so it can't be used for anything else at the same time. That and VPC 2007 doesn't work properly/at all on newer versions of Windows and many new computers don't have parallel ports, although if you do have one you could probably substitute another virtualisation platform as long as it supports parallel port passthrough and also can visualise a network card which has Windows 95 drivers available. Possibly you could also substitute Windows 98 in too, although I don't know if that was able to directly use DOS storage drivers the way that 95 was?