To find the smallest possible value of $xy$ given the equation $\frac{1}{x} + \frac{1}{2y} = \frac{1}{7}$, we can start by multiplying both sides of the equation by the least common multiple (LCM) of $x$ and $2y$, which is $2xy$. This will help us eliminate the denominators:
$2xy \cdot \left(\frac{1}{x} + \frac{1}{2y}\right) = 2xy \cdot \frac{1}{7}.$
Simplifying, we get:
$2y + x = \frac{2xy}{7}.$
Now, let's rearrange the equation to isolate $x$:
$x = \frac{2xy}{7} - 2y.$
Next, we can factor out $x$ on the right side:
$x = x \cdot \left(\frac{2y}{7}\right) - 2y.$
Dividing both sides of the equation by $x$, we have:
$1 = \frac{2y}{7} - \frac{2y}{x}.$
Simplifying further, we get:
$1 = \frac{2y}{7} \cdot \left(1 - \frac{2}{x}\right).$
Now, let's consider the equation $1 - \frac{2}{x}$. Since $x$ and $y$ are positive integers, the value of $\left(1 - \frac{2}{x}\right)$ must be positive. Therefore, $1 - \frac{2}{x} > 0$, which implies $\frac{2}{x} < 1$.
To minimize the value of $xy$, we want to find the smallest possible positive integer values for $x$ and $y$ that satisfy the given equation. Since $\frac{2}{x} < 1$, the smallest possible value for $x$ is 2.
Substituting $x = 2$ into the equation $1 = \frac{2y}{7} \cdot \left(1 - \frac{2}{x}\right)$, we have:
$1 = \frac{2y}{7} \cdot \left(1 - \frac{2}{2}\right) = \frac{2y}{7} \cdot 0 = 0.$
This implies that $y$ can take any positive integer value. Therefore, the smallest possible value of $xy$ is $x \cdot y = 2 \cdot 1 = 2$.
Hence, the smallest possible value of $xy$ is 2.