Judy, I think it would be doable, especially if you have access to a varied, commercially-available raw diet (Bravo comes to mind), and/or enough of a variety of RMB, muscle & organ found locally.
If you had the option for even a compact chest freezer (3-4 cu. ft) it would make storage easier. Something that size would work for a single dog, doesn't take up much space, and isn't as cumbersome to move as a standard deep-freeze. Additionally, it would allow you to take advantage of finding various foods on sale and storing them, if you can't find a bulk supplier in your area. It would also eliminate the issue of storing the dog's food next to the family's ice cream.
With a little bit of work & resourcefulness, you can feed a complete raw diet. The key points are to ensure a correct RMB/muscle/organ ratio, and to feed as much variety as you possibly can. This doesn't mean you have to go crazy feeding the more exotic stuff however, if you were limited to only one food source (i.e. just chicken) it would be necessary to take another approach to feeding.