Let $\mathcal{S}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abc}$ where $a, b, c$ are distinct digits.  Find the sum of the elements of $\mathcal{S}$.
1. We start by understanding the set $\mathcal{S}$, which consists of numbers that can be written as repeating decimals of the form $0.\overline{abc}$, where $a, b, c$ are distinct digits. Each number in $\mathcal{S}$ can be expressed as a fraction:
   \[
   0.\overline{abc} = \frac{abc}{999}
   \]
   where $abc$ is a three-digit number formed by the digits $a, b, c$.

2. The total number of such three-digit numbers $abc$ where $a, b, c$ are distinct digits is calculated as follows:
   - There are 10 choices for $a$ (0 through 9).
   - There are 9 remaining choices for $b$ (excluding the digit chosen for $a$).
   - There are 8 remaining choices for $c$ (excluding the digits chosen for $a$ and $b$).
   Therefore, the total number of distinct three-digit numbers $abc$ is:
   \[
   10 \times 9 \times 8 = 720
   \]

3. Next, we need to find the sum of all elements in $\mathcal{S}$. Notice that for each $k$ in the set $\{012, 013, \ldots, 986, 987\}$, there is a corresponding $(1000 - k)$ such that:
   \[
   \frac{k}{999} + \frac{1000 - k}{999} = \frac{1000}{999} = 1
   \]
   This means that each pair $(k, 1000 - k)$ sums to 1.

4. Since there are 720 elements in $\mathcal{S}$, there are $\frac{720}{2} = 360$ such pairs. Each pair sums to 1, so the total sum of all elements in $\mathcal{S}$ is:
   \[
   360 \times 1 = 360
   \]

The final answer is $\boxed{360}$.