To find the volume of the right rectangular prism, we first need to calculate the total volume occupied by the 24 smaller cubes.
Each cube has an edge length of \( \frac{1}{4} \) meter. The volume \( V \) of a single cube is given by the formula:
\[ V = \text{edge length}^3 \]
Substituting the edge length:
\[ V = \left(\frac{1}{4}\right)^3 = \frac{1}{4} \times \frac{1}{4} \times \frac{1}{4} = \frac{1}{64} \text{ cubic meters} \]
Now, since there are 24 cubes, the total volume of the prism \( V_{\text{total}} \) is:
\[ V_{\text{total}} = 24 \times \frac{1}{64} = \frac{24}{64} = \frac{3}{8} \text{ cubic meters} \]
Thus, the volume of the right rectangular prism is
\[ \boxed{\frac{3}{8}} \text{ cubic meters}. \]