Problems with an old 1930 dimmer switch wiring
I suspect the wire that is grounded was the hot feeding the light switch and it is shorted somewhere.
If there are only 2-wires in the box it would have to have been a hot and a switchleg. Since you have no power there I would have to guess the wire that has 5.4V is the switchleg and the other was the hot because there is no reason for a neutral to be in a switch box. I suggest you trace that wire, it may go to the light and be shorted there.
It is not uncommon in older homes wired with old style BX cable that has cloth covered rubber insulated wiring to short right where it enters the box. This is because they used shallow pancake type boxes for ceiling outlets and the wire is usually bent sharply where it enters. This was not a problem when the wiring was new but now the rubber is dried out and crumbling leaving the copper exposed. There is no reasonable and safe way to repair this situation other than to replace the affected cables. This quite often leads to replacement of the entire circuit/s because as you take apart one splice the insulation starts to crumble on all the wiring.
If you are lucky and the house is piped in you can repull but if it is BX you will have to cut and fish.
I have also seen in older homes where the neutral was switched, but you said a single pole switch didn't work so that is not the case.
Answers based on the mood I am in when you ask!