jmc

Because $\lfloor x\rfloor$ and $\lfloor y\rfloor$ are both integers, they must be a factor pair of $16,$ possibly both negative. For each factor pair of $16,$ say $(a,b),$ the graph of the equations $\lfloor x\rfloor =a$ and $\lfloor y\rfloor =b$ is a unit square (aligned with the axes) whose bottom-left corner is $(a,b),$ so the area of the graph is simply $1.$ Therefore, the area of the given region is equal to the number of factor pairs of $16.$

Since $16$ has $5$ positive factors (namely, $1,2,4,8,16$) and $5$ negative factors, there are $5+5=\boxed{10}$ ordered pairs of integers $(a,b)$ such that $ab=16,$ which is the answer.