Yes it looks like the battery has died.
I have a PS/2 Model 60 and they run MS-DOS 6.22 fine, no need for OS/2. They are lovely bits of kit, built like tanks. Well worth preserving.
The battery is an unusual camera-type battery which is mounted in a battery holder attached to the speaker which on the Model 50 is mounted front left. The PS/2 range was designed for tool-free maintenance with typical IBM engineering and attention to detail and it's simple to replace. No messing around with batteries soldered to the motherboard like many cheaper machines of the time.
You should then run the setup ("reference") disk which can be downloaded here-
http://www.walshcomptech.com/selectpccbbs/