How can I add a grounding conductor to my early 20th century electrical system?
In the US we need to conform to the NEC code. That code requires two copper ground rods 10' long to be driven into the ground with only a couple inches exposed. 


The rods must be placed a certain distance apart. 
The ground must also be connected to the main cold water copper pipe entering the house
The water meter must have a wire jumper across it of the same required gauge of the main ground
The main breaker panel must be equipped with an approved grounding bar and the breaker box "bonded".


I don't know the codes where you are, but I suggest you get an electrician that knows the local codes. Improper methodes of grounding can be dangerous and lethal!

Also I can tell you that your washing machine problem is not due to a missing ground. The machine should be grounded for safety, but it will not cause the problems you have.