I'm trying to find a set of uniform measure 1/2 over $ \{ -1,1 \} ^n \times \{-1,1\}^n$ such that the inner product of $(x,y)\in\{ -1,1 \} ^n \times \{-1,1\}^n$ will hold $|\langle x,y\rangle|< \frac{\sqrt(n)}{c}$ for some constant $c$.

I believe that a better way to look at it is saying I have a simple random walk. How do I find $r$ such that after $n$ steps the random walk will land in $(-r,r)$ w.p. 1/2 ? (Hopefully, for my needs and as I suspect, in fact $r$ will be $\Theta (\sqrt(n))$).

Formally, take independent random variables $Z_1, Z_2,\dots$, where each variable is either $1$ or $-1$, with a 50% probability for either value, and set $S_0 = 0$ and $S_n =\sum_{j=1}^nZ_j$. for what $r$ holds $$Pr[|\sum_{j=1}^nZ_j| \leq r] = 0.5$$