jmc

Multiplying through by $14xy$, we have $14y+7x=2xy$, so $2xy-7x-14y=0$. We then apply Simon's Favorite Factoring Trick by adding $49$ to both sides to get $2xy-7x-14y+49=49$. We can then factor this to get $$(x-7)(2y-7)=49$$Since $49$ factors to $7\cdot 7$ and $x$ and $y$ must be positive integers, the only possible solutions $(x,y)$ are $(8,28),(14,7),\text{and }(56,4)$. Out of these, $(14,7)$ yields the least possible value $xy$ of $\boxed{98}$.