jmc

Let $S$ denote the given sum. Then $$\begin{array}{rl}2S& =(a_1+a_2+\cdots +a_{100}{)}^2-(a_1^2+a_2^2+\cdots +a_{100}^2)\\ & =(a_1+a_2+\cdots +a_{100}{)}^2-100.\end{array}$$To find the minimum positive value of $2S,$ we want $(a_1+a_2+\cdots +a_{100}{)}^2$ to be as close to 100 as possible (while being greater than 100). Since each $a_i$ is $1$ or $-1,$ $a_1+a_2+\cdots +a_{100}$ must be an even integer. Thus, the smallest we could make $(a_1+a_2+\cdots +a_{100}{)}^2$ is $12^2=144.$ This is achievable by setting 56 of the $a_i$ to be equal to $1,$ and the remaining 44 to be equal to $-1.$

Thus, the minimum positive value of $S$ is $\frac{144-100}{2}=\boxed{22}.$