I am building a passive solar house in Lake Tahoe in the Angora burn area. So there are no trees, a pure south face and we have a six inch insulated slab in the living room. I am a forester, so I get lots of free fire wood. Tahoe also gets about 300 days of sun per year. I am concerned that my slab may get cold when we do get a week of cloudy weather. I dont want to pay for hydronic through the house because I only pay $30 per month to heat a very poorly constructed house, the new house is going to be very tight. What can I do to backup heat my slab during those rare periods when we get no sun?

You first have to know your heat load. Solar is not all about sun, but rather how much energy you need to satisfy the load. If your load is low and sun abundant, you may use electric boiler backup. Few environments in the US will match solar with space heating at a reasonable cost, save the southwest.

Hmm, an electric boiler could do the trick. I am only looking to heat the slab for about a month of the year. This year is an el nino and we are having more cloudy days than usual. So my slab would be cold by now. Tahoe does resemble the southwest in sunny days. And the slab is really only there to help, we have forced air, wood stove and a slab. I just want to be able to keep the slab warm during those rare periods when we have consistent overcast. I need to be able to do that on a budget. We have a perfect place by the slab to put a small boiler. The slab is 20x18 so it is not a huge area.

Sounds like you need either electric heating wires or hydronic tubes in the slab. Or just let it go - it won't get that cold if the slab is insulated and the air above it is fully heated.

20x18x25 BTU= 9,000 nice little electric boiler would work well and provide back up heat if your other system fails. You would also have surplus capacity if you were to hit another zone say the bath rooms, options.
The smallest we have is about 20,000. But it modulates down and is a nice fit. Electric boilers are a straight forward install, and very reliable.
Dan