That sounds like a very odd setup, and certainly sounds like it will need rewiring.
When I moved into my house all the electrical cable was lead-coated, with rubber insulation around the live and neutral. It still had an earth though despite being 80 years old.
It may be that the neutral and earth are linked instead. After all, neutral is just a massive earth rod at the substation. Massive bodge way of doing it, but the effect should be the same.
You could see if this is the case by testing for continuity between earth and neutral by (obviously) turning off the supply entirely (not just the circuit you're testing it on), set a multi-meter to measure resistance and see if you get continuity when you touch one lead to the bare end of an earth cable, and the other to a bare neutral (having obviously used an electricians screwdriver to make sure there is no voltage on either first for some reason!)
Everything in the house will need to be unplugged though as you could get continuity through an appliance somewhere. If you get continuity then the neutral is doubling as the earth. If you don't... well, it's not.
By the way I'm not an electrician. If you blow yourself up it's your lookout!
