$n$ is a positive even divisor of $1000$ if and only if $n = 2m$ where $m$ is a divisor of $500$. Since $500 = 2^2 \times 5^3$, there are $(2+1)(3+1) = 12$ divisors of $500$. Those divisors are

\begin{array}{rr}
1, & 500, \\
2, & 250, \\
4, & 125, \\
5, & 100, \\
10, & 50, \\
20, & 25 \\
\end{array}

so there are $12$ positive even divisors of $1000$.

Those divisors are

\begin{array}{cc}
2, & 1000, \\
4, & 500, \\
8, & 250, \\
10, & 200, \\
20, & 100, \\
40, & 50 \\
\end{array}

The sum of the positive divisors of $500 = 2^2 \times 5^3$ equals
$\dfrac{2^3 - 1}{2 - 1} \times \dfrac{5^4 - 1}{5 - 1} = 1092$

So the sum of the even divisors of $1000$ is $2 \times 1092 = 2184$