Chances are you have a fault. If it never resets, it's almost assuredly a ground-to-neutral fault somewhere. First step is to disconnect the downstream outlets and see if the GFCI resets. If so the fault is downstream. You can reconnect it and then go and disconnect the third outlet at the second, etc... to isolate the problem. Once you've found it look carefully for the bare ground touching the neutral screws or the neutral wire touching something (the box or one of the ground wires).

OK. I disconnected the black & white wires on the LOAD side of the GFCI and got the same results - 120V across the LINE/GROUND wires but nothing in the GFCI slots. The wiring into the GFCI appears to be tight.

What about LINE-NEUTRAL?

Across the hot and neutral it's measuring 5.6V.

Something's wrong with your feed. You should get 120V HOT-NEUTRAL.

I thought the GFI outlet was first in that circuit. Is it possible there's something else before it or is just it bad wiring between the panel and the outlet? Nothing's ever been changed - it just happened.

If you have power from hot to ground but you are reading nothing or very low voltage from hot to neutral then you have a lost neutral somewhere. Look for a j-box or maybe the panel it comes from, worse case scenario you need to run a new wire and abandon the old one.

I concur with nick on this one. Obviously the hot and most likely the ground are intact. The neutral isn't. As far as your meter is concerned, neutral and ground should be synonymous.

Thanks guys, you've given me something to research. If I can't find anything, I'll call in a professional.