To construct a segment equal to the cross-ratio, you slightly modify the first picture in
the Wikipedia article. Let us call $P$ the point in the top where the 4 black lines cross.
Suppose you want to construct two segments whose ratio equals
to the cross-ratio of 4 given points on a line.
Referring to this picture, suppose that your given points are $A',B',C',D'$, and they are
on the red line $L'$. Then draw a second red line L, and choose the point $P$ so that
$PD'$ is parallel to $L$. (Or choose $P$ arbitrarily, and then choose $L$ parallel to
$PD$). Then use this $P$ as the center of projection and
project $A',B',C'$ on $L$ from $P$. The ratio $AC/BC$ will be equal to the cross ratio
To obtain the "numerical" value (the notion foreign to the Greek mathematics;
after the discovery of non-commensurable segments they did not measure the segments
with numbers; they were only speaking of proportions, see Euclid), you have to choose the unit of length.
Suppose that $BC$ has length $1$,
then $AC$ is the cross ratio.
I have not checked with Pappus, but I suppose he had some similar construction, if he had any.