I've heard this a few times and I think there are a couple of things that always come to mind. It depends on what kind of rabbit your talking about. A farm raised pasture fed meat rabbit will have a nice bit of fat on it. It's very light fat with little flavor but there are a few ounces of it on each rabbit. This assumes that you are raising them with a good feed source and butchering them at around 8-10 weeks old. Older rabbits will have more fat.
If you're dealing with wild rabbits such as cotton tails or snowshoe hares, you'll not likely find much fat. Mostly because these rabbits are more active and they have to keep moving to stay ahead of predators. It's important for rabbits to not carry much extra fat for them to breed successfully. Fat rabbits have smaller litters, less frequently and the males are less likely to want to mate if they are too fat. So it seems that it is in their nature to live thin to be productive.
As for starvation, you'll probably starve slower with rabbits than without them. Food is food at that point, right? If you're in a position where rabbits are all that you have left to eat, I'd suggest eating as much of that critter as you can. Heart, lungs, liver, kidney, eyes, bones. You can pretty much eat all of it, except the hair and teeth. If you're out hunting wild rabbits to keep from starving to death, I'd also suggest carrying a few field guides for wild plants, bugs and fungus that you might find along your way. There's many insects that have a good fat profile and that don't taste all that bad. Of course, if your starving, and hunting for food, there's no such thing as a hunting season, so don't pass up opportunities for trapping small birds, mice, squirrels, chipmunks and other wild edibles if it means staying alive.
I really like rabbits. They are my favorite farm critter, because they are quiet, productive and easy to care for. Like anything else though, they can't be relied on as your only protein and fat source. I think a good balance for a rabbitry is to have a breeding pair of pigs to provide nutritionally rich meat and a few extra pounds of fat. There's always more fat on a pig than most people like, so I save the extra, for mixing in with rabbit meat and making sausage. They work very well together and nobody starves.