The sum of the two perfect cubes that are closest to $500$ is $343+512=855$.  Find the sum of the two perfect cubes that are closest to $2008$.
1. To find the sum of the two perfect cubes that are closest to $2008$, we first need to identify the perfect cubes around this number.
2. We start by finding the cube roots of numbers around $2008$:
   \[
   \sqrt[3]{2008} \approx 12.6
   \]
3. We then check the cubes of the integers immediately below and above this value:
   \[
   12^3 = 12 \times 12 \times 12 = 1728
   \]
   \[
   13^3 = 13 \times 13 \times 13 = 2197
   \]
4. We see that $1728$ and $2197$ are the perfect cubes closest to $2008$.
5. Adding these two cubes together:
   \[
   1728 + 2197 = 3925
   \]

The final answer is $\boxed{3925}$