  Superconductivity in the iron pnictides emerges from metallic parent
compounds exhibiting intertwined stripe-type magnetic order and nematic order,
with itinerant electrons suggested to be essential for both. Here we use X-ray
and neutron scattering to show that a similar intertwined state is realized in
semiconducting KFe$_{0.8}$Ag$_{1.2}$Te$_2$ (K$_5$Fe$_4$Ag$_6$Te$_{10}$) without
itinerant electrons. We find Fe atoms in KFe$_{0.8}$Ag$_{1.2}$Te$_2$ form
isolated $2\times2$ blocks, separated by nonmagnetic Ag atoms. Long-range
magnetic order sets in below $T_{\rm N}\approx35$ K, with magnetic moments
within the $2\times2$ Fe blocks ordering into the stripe-type configuration. A
nematic order accompanies the magnetic transition, manifest as a structural
distortion that breaks the fourfold rotational symmetry of the lattice. The
nematic orders in KFe$_{0.8}$Ag$_{1.2}$Te$_2$ and iron pnictide parent
compounds are similar in magnitude and how they relate to the magnetic order,
indicating a common origin. Since KFe$_{0.8}$Ag$_{1.2}$Te$_2$ is a
semiconductor without itinerant electrons, this indicates that local-moment
magnetic interactions are integral to its magnetic and nematic orders, and such
interactions may play a key role in iron-based superconductivity.
