If the app is stable running under Windows 95, a viable approach could be to run W95 and the app stack in a VM under a modern and properly secured hypervisor. The host system provides security and stability, and 95 gets all the resources it needs in a sandboxed environment. It would probably run faster than it currently does directly on old hardware. It would certainly be safer.
As dangerous as it is to run on an obsolete OS, there are risks in migration as well, especially with a service that needs near constant availability. A VM could allow this obviously reliable app to run safely without changes for years to come and to migrate seamlessly to new hardware again when the time comes.
As dangerous as it is to run on an obsolete OS, there are risks in migration as well, especially with a service that needs near constant availability. A VM could allow this obviously reliable app to run safely without changes for years to come and to migrate seamlessly to new hardware again when the time comes.