There is a tiny watch size battery cell on power board, top right side (view from LCD side) under top cover. Little challenge to replace as it's really small and It is welded to a two flat steel lines (not soldered). Not sure the voltage as mine is also dead and shows 0.6v and it has no any markings on it. Could be around 1.5v but can also be a 3v. But seems that there is an extra room around for a larger size cell may fit.

You may want to check and replace a fuse on 5D power board (top board on the right size). That fuse could be one of the reasons. There is a video published on youtube except that it’s for 5D Mk2 and the diff is that power board of MkII has many fuses but 5D classic has only one and no need to remove power board to replace that fuse. The only hope that there are no other issues behind as fuses don’t burn without the reason otherwise it’ll be a longer story. The fuse type is Littelfuse SMD SMT 0603 Fast Acting Fuse 1.5A 32V 046701.5 or simply ”Littelfuse 0603 Type K” where K stands for 1.5A. And make sure you have basic micro soldering skills otherwise would be cheaper to ask for someone’s help..