It is best to wear as much as you can when it gets really cold without any electricity. Try to wear a sweater and a jacket if it is very cold. Woolen products will certainly help since they trap body heat quite well. It will make you feel a touch heavy but the comfort it will bring will certainly outweigh the slight discomfort.
If you have a fireplace, you can certainly make use of that. It is not a bad idea to have one in areas where the weather is severely cold during the winter. Very few are operated with electricity and most of them are either gas or firewood based. These provide ample heat for the room in which they are placed, even in the coldest of weather. It also is a great way of getting the family together in one place.
An alternative option is the use of gas based heating especially in areas where the bad weather might knock out the electricity. It is also not a bad idea to have at least one device that can be connected with gas and used in adverse conditions should your central heating be connected to electricity.
If you have the space for a bonfire, it is a really fun option. A bonfire can provide a lot of warmth and light when the electricity is out. You can have a party going with friends and family in the outdoors. A bit of a barbecue can also add even more enjoyment. The lack of electricity can bring the whole family together.