I recently rescued a 486 mobo that had some bad Varta leakage. Luckily the corrosive goop hadn't destroyed any important traces, because the battery was mounted in an area of the board that didn't really have anything nearby. It's important to neutralize the gunk (I used vinegar) and then rinse the area thoroughly with isopropyl alcohol. That will stop any further damage, but depending on how bad the leakage was, some of the PCB traces might've been destroyed.
But that might not be your issue. Check for any blown tantalum capacitors (they're the bright orange/yellow components) -- I had one of them go pop in that 486 board. When they do go, they'll often short out one of the power rails, at which point the PSU will usually shut down. Check the resistance at the motherboard power connectors. If any of the voltage rails (+5V, -5V, +12V, -12V) have an usually low resistance -- say, 100 ohms or less -- then you've got a short somewhere on the board, and 9 times out of 10 that will be due to a tantalum.