jmc

We begin by factoring $336$. $336=2^4\cdot 3\cdot 7.$ Because we are looking for phone numbers, we want four single digits that will multiply to equal $336.$ Notice that $7$ cannot be multiplied by anything, because $7\cdot 2$ is $14,$ which is already two digits. So, one of our digits is necessarily $7.$ The factor $3$ can be multiplied by at most $2,$ and the highest power of $2$ that we can have is $2^3=8.$ Using these observations, it is fairly simple to come up with the following list of groups of digits whose product is $336:$ $$\begin{array}{rl}& 1,6,7,8\\ & 2,4,6,7\\ & 2,3,7,8\\ & 3,4,4,7\end{array}$$For the first three groups, there are $4!=24$ possible rearrangements of the digits. For the last group, $4$ is repeated twice, so we must divide by $2$ to avoid overcounting, so there are $\frac{4!}{2}=12$ possible rearrangements of the digits. Thus, there are $3\cdot 24+12=\boxed{84}$ possible phone numbers that can be constructed to have this property.