I don't see MS-DOS or Windows 3.11 as much as a problem as having 30 year old hardware. Bad capacitors on the motherboard and oxidated (rusty) contacts for memory and ISA cards can break the system, while the software itself can be ran in a virtual machine on newer hardware, if needed.

I ran MS-DOS in an airport for several years and it was just fine. Windows 2000 was available, but not needed for the task at hand, while a reboot was way faster with MS-DOS than any Windows at that time and a reset was the solution for 99% of the incidents, the rest was hardware-related.

In terms of replacements, a Raspberry Pi would be more than enough in terms of computing power, with some decent (non-microSD) local storage. Cheap, simple and I guess the cost of writing the MS-DOS app from scratch should cost about the same as that admin for 1 year.