No, you cannot bake a full-size cake in a microwave alone. Baking a cake requires a conventional oven that surrounds the cake with dry, heated air, allowing it to cook evenly from all sides. Microwaves, on the other hand, work by emitting electromagnetic waves that primarily heat the water molecules present in food.
While you can make certain microwave cake recipes that are specifically designed for microwave cooking, these are usually smaller, single-serving cakes and have a different texture compared to traditional oven-baked cakes. They are also typically simpler recipes that rely on the microwave's quick cooking time.
To bake a full-size cake with the desired texture and consistency, you will need a regular oven. Baking a cake in a traditional oven ensures that it bakes evenly, rises properly, and develops the desired crust and crumb.
If you don't have access to a conventional oven, you may want to consider alternatives like a toaster oven or an electric convection oven, which can serve as smaller, portable options for baking cakes and other baked goods.