hub bearings won’t stay in place

very amateur bike mechanic here, currently trying to fix up my girlfriend’s old bike. the issue is:
the rear wheel keeps wobbling around! after trying for days to find out what’s wrong i realised one side of the rear hubs bearings just won’t stay in place, causing the wheels axle to shift back and forth.
the axle is not bent and looks fine, the hubs are from weco (presumably very old)
anybody have experience working old bikes?
thanks so much!

hub bearings won’t stay in place
byu/_some_dude__ inbikewrench

by _some_dude__