Label the square as $A B C D$. Suppose that the point $X$ is 1 unit from side $A B$. Then $X$ lies on a line segment $Y Z$ that is 1 unit below side $A B$. Note that if $X$ lies on $Y Z$, then it is automatically 4 units from side $D C$. Since $X$ must be 2 units from either side $A D$ or side $B C$, then there are 2 possible locations for $X$ on this line segment. Note that in either case, $X$ is 3 units from the fourth side, so the four distances are 1, 2, 3, 4 as required. We can repeat the process with $X$ being 2,3 or 4 units away from side $A B$. In each case, there will be 2 possible locations for $X$. Overall, there are $4(2)=8$ possible locations for $X$. These 8 locations are all different, since there are 2 different points on each of 4 parallel lines.