  Let $K$ be a field of characteristic $\ne 2$ and let $X$ be the affine
variety over $K$ defined by the equation $$ X:\ a_0x_0^2 + \cdots + a_nx_n^2 =
1 $$ where $n\ge 0$ and $a_i\in K$. In this paper we compute the lowest mod 2
\'{e}tale homological obstruction class to the existence of a $K$-rational
point on $X$, and show that it is the cup product of the form $$ o_{n+1} =
[a_0]\cup\cdots\cup[a_n]. $$
  Our computation is an \'{e}tale-homotopy analogue of the topological fact
that Stiefel-Whitney classes are the homological obstructions to find a section
to the unit sphere bundle of a real vector bundle.
