Is a boxed-mix cake sturdy enough to construct a sphere?
There are a number of ways to increase the density of a box-cake mix.  Add the number of eggs in the direction, but add two extra egg yolks. Instead of the water, use the same amount of whole milk. 

Another method is to add a package of instant pudding, and many mixes have a recipe for this right on the box. To each box mix, use a 3.4 ounce package of instant pudding, 4 eggs, 1 cup water (or milk) and reduce the amount of vegetable oil to 1/3 cup.

And one I found years ago, who posts on Cake Central as doreen1092, offered this: to each box cake mix, add 1/2 cup sour cream, 2 eggs, 1/4 cup sugar, and 8 ounces cream cheese. I beat the cream cheese, then add the eggs and sugar, the sour cream and, lastly, the mix.