Suppose that $x_i$'s exist. Let $a_i:=x_ix_{i+1}$ (where $x_{2010}=x_1$). We see that $1504$ of $a_i$'s must be $+1$, while $505$ of them are $-1$. That is, \[+1=\prod_{i=1}^{2009}x_i^2=\prod_{i=1}^{2009} a_i = (+1)^{1504}(-1)^{505} = -1\,,\] a contradiction.

Notice that the amount of $x_ix_{i+1}$'s is $(2009)$ ; $x_ix_{i+1}$ can be $1$ or $-1$, let's separate $L$ as the set of $x_ix_{i+1}$'s that are equal to $-1$ and $M$ the set of $x_ix_{i+1}$'s that are equal to $1$ and let $N$ the set of all $x_ix_{i+1}$'s. Now we have that $|M| - |L| = 999$. $|M| + |L| = 2009$. so $|M| = 1504$ and $|L|=505$ Let $ F(W)$ be the product of the elements of the set $W$. So, $F(M)F(L) = (1^{1504})(-1)^{505} = -1 = F(N) = x_1^2x_2^2...x_{2009}^2 = 1$. Contradiction. NOTE: $|H|$ represents the number of elements in the set $H$