Yes, but it's a pain (and depends how much kitchenware you have). When I was in the Peace Corps, I learned how to construct an "oven". You'll need two pots of the same diameter that are big enough to enclose your cake pan, hopefully with about at least an inch of room on all sides. (Just one pot is needed if it's tall enough.)
Put one pot on the burner, then put something in it to elevate the cake pan at least 1/2 inch above the bottom of the pot. Three small rocks work well, as will three stacks of coins. Put the cake pan on top of this stand, then put the last pot upside-down on top as a lid. (If your first pot is tall enough, you would only need a regular lid, not a second pot.) It's best to set this up on the burner as moving it can be risky. Use a low setting for the burner and expect a longer baking time.
This method can work on a stove or on a charcoal grill (with very low heat in the case of grill). It will work better if you can add heat from the top as well (e.g. put a few coals on top). It's finicky though and takes some practice. Bread is a pretty safe thing to try, though I would guess that a box-cake would do okay.
Microwave cake is an easier option, just google "microwave cake" and you'll find lots of recipes.
There are lots of people who have a rather simplistic approach to nutrition and think that removing fat and calories makes you healthy. Then they go through recipes for things they want to eat, replace the sources of fat with something which doesn't have fat and doesn't make the result outright inedible, and declare their recipe a success. I think this is what happened here.
In a cake, eggs provide leavening, moisture, smoothness, own flavor, and enhancement of other flavors. Oil provides smoothness and enhancement of other flavors (and possibly its own flavor, if not netural). And while it is not water based, it keeps the moisture in the cake from evaporating, so it makes the cake less dry.
If you are a "simplistic nutritionist" without all this information, you can approximate some of the effects with soda. It will provide moisture, and it will also provide some leavening because it is fizzy. It will provide some flavor of its own too, but frankly, I find the rather chemical flavor of soda to be unpleasant. And it won't have any fat. In the eyes of the simplistic nutritionist, it has successfully replaced the oil and eggs while reducing fat and calories.
From the point of view of a baker, the cake will be a disaster, and won't even deserve the label cake. It will dry out quickly because it has no fat. It will have a bland flavor. Its texture will be terrible. They say "more chewy?" It will miss both the protein structure and the emulsifying agents provided by the eggs. It will be essentially an overwhelmingly sweet quickbread with no redeeming qualities. From a culinary point of view, it will be terrible.
Bottom line: under some assumptions, it is a good substitution. For me, these assumptions are so far from reality as to be useless. It is a terrible substitution.
Best Answer
You can "steam" a cake in the pressure cooker using the Bain Marie or Pan-in-pot method. Basically put the cake batter in a smaller heat-proof container (usually 7" or 20cm) that is buttered as usual. Then in the pressure cooker add enough water to equal the minimum liquid (usually 1-2 cups), and a steamer basket or rack. Add the container containing the batter onto the rack and pressure cook for about 20 minutes with natural release. You'll want to increase the baking soda or powder by 50% to compensate for the extra pressure - but generally most of the "rise" in the cake will happen while the cooker is reaching pressure.
P.S. Be sure to construct a sling out of foil (a strip folded three times and placed under the container with the ends coming up the side of the cooker) to more easily lower and raise the cake.