  In this paper we describe the birational geometry of Fano double spaces
$V\stackrel{\sigma}{\to}{\mathbb P}^{M+1}$ of index 2 and dimension $\geqslant
8$ with at mostquadratic singularities of rank $\geqslant 8$, satisfying
certain additional conditions of general position: we prove that these
varieties have no structures of a rationally connected fibre space over a base
of dimension $\geqslant 2$, that every birational map $\chi\colon
V\dashrightarrow V'$ onto the total space of a Mori fibre space $V'/{\mathbb
P}^1$ induces an isomorphism $V^+\cong V'$ of the blow up $V^+$ of the variety
$V$ along $\sigma^{-1}(P)$, where $P\subset {\mathbb P}^{M+1}$ is a linear
subspace of codimension 2, and that every birational map of the variety $V$
onto a Fano variety $V'$ with ${\mathbb Q}$-factorial terminal singularities
and Picard number 1 is an isomorphism. We give an explicit lower estimate for
the codimension of the set of varieties $V$ with worse singularities or not
satisfying the conditions of general position, quadratic in $M$. The proof
makes use of the method of maximal singularities and the improved
$4n^2$-inequality for the self-intersection of a mobile linear system.
