The internal battery going sounds likely Ina .
It may be a button one or it could be like a short AA or AAA battery on the motherboard .
It powers the basic start up procedure , the cmos memory that dates back to the beginning of pc's .
Some mother boards have easy to get at ones but all the ones i have had are soldered in position .
As long as the pc has power going into it , you can usually get away with this , but pull the plug out and then try to reboot when you put power back on and the cmos basic start up can't be accessed and at first things like the date go wrong , but you may end up with a black screen and the pc speaker going beeeep bip bip , and then nothing .
It is possible to change the battery , i have used watch batteries even AA ones , but it may be time for a new m/board or another second hand pc . You should be able to take out the hard drive of the present pc and put it in a new s/h one as the master or slave h/d , a 5 minute swap job .
Save any info you want asap just in case , save to memory stick or card via usb/reader .