In an old 60 amp box, there likely was never any "bare ground wire". The safety ground was not adopted at the time that box was installed.
I have the same problem. My house was built around 1960.
Basically nothing to do but replace all the wiring, if you want the ground.

If it`s a small single story house, replacing all of the outlet wiring probably won`t be that big a deal, I`ve done it. Just remember that you probably also have metal outlet boxes and they must be bonded to the ground wire at each location but your wiring will be exponentially safer.

A cheaper solution is to install GFCI's in the exsisting receptacles....it'll save a lot of work,that's for sure...........take a look at your wires,to see if they are cloth covered,or plastic covered(sheathed).Also note the size of the wire,it could be #12,but most likely,it's#14...........if you plan on updating any appliances,or adding on,in the future,your best bet would probably be to re-wire..........