The challenge is the fact that there are so many voltage rails in a standard PC vs a notebook so if you want to do it on the DC side it would be messy. There's even negative voltage rails to deal with. So you almost need a stage in between, but that would be high voltage most likely. Ex: at the filter capacitors you could have a string of batteries that float. But we're talking like 170VDC. That stage is also not regulated, which you would need for proper float. (2.25v per cell)
Easiest bet might be to try to integrate a UPS like circuit into the case. AC in to the case-> UPS inside case -> AC out -> PSU. You'd most likely want to rewire the UPSes on/off switch to be outside the case too.
Though there might possibly be some motherboards on the market that only take a single DC input voltage like 12 volts, but those would be hard to find and greatly limit your selection. But if you can find such motherboard then you could actually have a 13.5v AC to DC PSU that powers it, and then stick a lead acid battery in parallel. AT 13.5v it will keep the battery charged and shoudl be within spec for a 12v motherboard (you'd want to check this though). You would want an Arduino or something to detect voltage so that it shuts down if it gets too low, or else it would completely kill the battery. You could have a relay simply toggle the power button as it should initiate a proper shut down of the system. so say, at 11.3 volts it could initiate shutdown, at 11.0 it would actually cut power completely (in case the shutdown didn't actually work, ex: os frozen).
Another option might be to try to find a 48vdc PSU but those are hard to find too.