To solve this question, we first need to understand how gravitational force (and hence weight) changes with depth under the surface of the Earth. The gravitational force at a depth $d$ is given by the formula:

$$ F = F_0 \left(1 - \frac{d}{R}\right) $$

where $F_0$ is the gravitational force (or the weight) at the surface, $R$ is the radius of the Earth, and $d$ is the depth below the Earth's surface.

In your question, the body weighs 300 N on the surface, so $F_0 = 300$ N. It is taken to a depth of $\frac{R}{4}$ under the surface. Therefore, $d = \frac{R}{4}$.

Substituting these values into our formula, we get:

$$ F = 300 \left(1 - \frac{\frac{R}{4}}{R}\right) $$

$$ F = 300 \left(1 - \frac{1}{4}\right) $$

$$ F = 300 \left(\frac{3}{4}\right) $$

$$ F = 225 \, \text{N} $$

Therefore, at a depth of $\frac{R}{4}$ under the surface of the Earth, the body would weigh 225 N. The correct option is Option D.