Can someone be genetically really skinny or really fat?

Yes and no: Calories are an absolute necessity to generate fat, and genetics can't trump the laws of physics but it can predispose people to storing rather than burning more calories. Aside from diet and exercise, many factors affect weight including one's size as a child or even as a newborn, levels of anxiety or fidgety behavior, medications, height, age, family (genetics), and even local climate.

Genes play a role: Genetics play a large role in health. Some families have people who are very skinny & some have members who are very overweight. However, diet & lifestyle play a role too. Those in the same families tend to have similar diets & lifestyles. In many cases we have 2 different versions of the same gene & diet & lifestyle affect which gene is expressed. So genes play a role but aren't the whole story.