  We introduce a new approach to the construction of sheaf theory on discrete
and combinatorial spaces by constructing sheaves on \v{C}ech closure spaces, a
category which contains the major classes of spaces of most interest to
applications: topological spaces, graphs, and metric spaces with a priviledged
scale. In the process, we construct a new Grothendieck topology on \v{C}ech
closure spaces, and we show that sheaves on \{C}ech closure spaces are
isomorphic to the sheaf of sections of an appropriately defined \'etale space.
