  Fix an integer $r\geq 3$. We consider metric spaces on $n$ points such that
the distance between any two points lies in $\{1,..., r\}$. Our main result
describes their approximate structure for large $n$. As a consequence, we show
that the number of these metric spaces is $\lceil \frac{r+1}{2}\rceil
^{{n\choose 2} + o(n^2)}$. Related results in the continuous setting have
recently been proved by Kozma, Meyerovitch, Peled, and Samotij. When $r$ is
even, our structural characterization is more precise, and implies that almost
all such metric spaces have all distances at least $r/2$. As an easy
consequence, when $r$ is even we improve the error term above from $o(n^2)$ to
$o(1)$, and also show a labeled first-order $0$-$1$ law in the language
$\mathcal{L}_r$, consisting of $r$ binary relations, one for each element of
$[r]$. In particular, we show the almost sure theory $T$ is the theory of the
Fra\"{i}ss\'{e} limit of the class of all finite simple complete edge-colored
graphs with edge colors in $\{r/2,..., r\}$. Our work can be viewed as an
extension of a long line of research in extremal combinatorics to the colored
setting, as well as an addition to the collection of known structures that
admit logical $0$-$1$ laws.
