  In this paper we investigate the moduli spaces of semistable coherent sheaves
of rank two on the projective space $\mathbb{P}^3$ and the following rational
Fano manifolds of the main series - the three-dimensional quadric $X_2$, the
intersection of two 4-dimensional quadrics $X_4$ and the Fano manifold $X_5$ of
degree 5. For the quadric $X_2$, the boundedness of the third Chern class $c_3$
of rank two semistable objects in $\mathrm{D}^b(X_2)$, including sheaves, is
proved. An explicit description is given of all the moduli spaces of semistable
sheaves of rank two on $X_2$, including reflexive ones, with a maximal third
class $c_3\ge0$. These spaces turn out to be irreducible smooth rational
manifolds in all cases, except for the following two: $(c_1,c_2,c_3)=(0,2,2)$
or (0,4,8). The first example of a disconnected module space of semistable rank
two sheaves with fixed Chern classes on a smooth projective variety is found --
this is the second of these exceptional cases $(c_1,c_2,c_3)= (0,4,8)$ on the
quadric $X_2$. Several new infinite series of rational components of the moduli
spaces of semistable sheaves of rank two on $\mathbb{P}^3$, $X_2$, $X_4$ and
$X_5$ are constructed, as well as a new infinite series of irrational
components on $X_4$. The boundedness of the class $c_3$ is proved for $c_1=0$
and any $c_2>0$ for stable reflexive sheaves of general type on manifolds $X_4$
and $X_5$.
