In plane geometry is it possible to represent the product of two line segments (p and q) as a line segment? In plane geometry the product of two line segments p and q can be represented as the area of a rectangle with sides p and q.  Or at least that is the premise assumed here. Assuming that is correct, the question is: Can this product be represented by a simple line segment, instead of an area?

Yes it is possible. 
We are given two line segments,  of lengths $a$ and $b$. Draw two say perpendicular lines (it doesn't really matter) meeting at some point $O$. On one of the lines, which we call the $x$-axis, make a point $A$ such that $OA=a$. (Straightedge and compass can do this.) On the other  line, which we call the $y$-axis, make a point $B$ such that $OB=b$.
On the $x$-axis, put a point $X$ such that $OX$ has unit length. Join $X$ and $B$. 
Through $A$, draw the line parallel to $XB$. This meets the $y$-axis at some point $P$.
By similar triangles, we have $\frac{b}{1}=\frac{OP}{a}$. It follows that $OP$ has length $ab$. 
Remark: By a small modification of the basic idea, we can also construct a line segment of length $\dfrac{a}{b}$.
Note that we need to define, perhaps arbitrarily, some line segment as the unit line segment in order to carry out the construction. 
