Is $\Gamma \vdash x x : T$ possible in the simply typed lambda calculus?
You are on the right track. The argument you would use is on the lines of 
size of types defined below: (I am assuming you are in the world of simply typed $\lambda$-calculus)

$size(T) = 1$

$size(T \to T') = size(T) + size(T')$

Unification will only work if the size of types is equal, and in this case 
$size(T \to T') &gt; size(T)$ hence there cannot be such a term.

However, if you add recursion/non-termination in the type system. 
You can indeed have such a term. 
$ \vdash (\lambda x. x x) (\lambda x. x x):\bot$ 

$\bot$ is the type that represents non-termination.