jmc

If $\frac{3}{11}=0.ababab\dots$, then, by multiplying both forms of this number by 100, we get $\frac{300}{11}=ab.ababab\dots$. Now we can subtract:

\begin{array}{r r c r@{}l} &300/11 &=& ab&.ababab\ldots \\ - &3/11 &=& 0&.ababab\ldots \\ \hline &297/11 &=& ab & \end{array}

We can simplify $\frac{297}{11}$ to $27$, giving us the two digits we sought: $a=2$ and $b=7$. Thus, $a+b=2+7=\boxed{9}$.

(Alternatively, we could solve this problem by long division.)