jmc

Note that $\gcd (a,b)$ divides both $a$ and $b$, so $\gcd (a,b)$ must also divide $a+b=1001$. Clearly, $\gcd (a,b)$ cannot be equal to 1001 (because both $a$ and $b$ must be less than 1001). The next-largest divisor of 1001 is 143. If $a=143$ and $b=1001-143=858$, then $\gcd (a,b)=\gcd (143,858)=143$. Therefore, the largest possible value of $\gcd (a,b)$ is $\boxed{143}$.