Sue,
The house I grew up in had a root cellar off the basement. It's ceiling was about 3-4 feet under ground, the sides had a layer of cement and the floor was dirt. This was in Minnestoa and during the summer it stayed quite cool (dad would store worms in there, mom would store fresh fruit and veggies). In the winter mom would store all here flower bulbs along with any fruit/veggies. It worked well, the temperature really did stay fairly constant, about 60-65F year-round.
Here in Texas there's a "living history" farm nearby and they have root cellar like what you are thinking of. They dug down into the earth about four feet, built a wooden room, then piled all the dirt back on top, along with even more dirt to make walls over five feet thick. The whole thing is about 6' tall by 10' long by 7' wide. I've been in it in hot weather and it might be 10F cooler which still puts it in the 80s. It's better than nothing but not by much.
Conditions in your area may not be as hot and you'll get better results. Sidenote: make sure you have good drainage or a sump pump to keep the water out.
-Blast