  A positroid is an ordered matroid realizable by a real matrix with all
nonnegative maximal minors. Postnikov gave a map from ordered matroids to
Grassmann necklaces, for which there is a unique positroid in each fiber of the
map. Here, we give forbidden minor characterizations of ternary and quaternary
positroids. We show that a positroid is ternary if and only if it is
near-regular, and that all ternary positroids are formed by direct sums and
$2$-sums of binary positroids and positroid ordered whirls. We prove that a
positroid is quaternary if and only if it is $U^2_6, U^4_6,$ and $P_6$-free.
Under the map from ordered matroids to Grassmann necklaces, we fully
characterize the fibers of ternary positroids, referred to as their positroid
envelope classes; in particular, the envelope class of a positroid ordered
whirl of rank-$r$ contains exactly four matroids.
