Answer: $4.418$

Explanation:1. Understand the shape of the box: Since each edge has the same length ($94$ cm) and each side is a square, the box is a cube. A cube has $6$ faces if it's closed. This box does not have a lid, meaning it has $5$ exterior square faces.

2. Calculate the area of one square face in square centimeters: Side length of a square face = $94$ cm. Area of one face = $(\text{side length})^2 = (94 \text{ cm})^2$. $94^2 = (90+4)^2 = 90^2 + 2(90)(4) + 4^2 = 8100 + 720 + 16 = 8836 \text{ cm}^2$.

3. Calculate the total exterior surface area of the box without a lid in square centimeters: The box has $5$ faces (bottom + $4$ sides). Total surface area = $5 \times \text{Area of one face}$. Total surface area = $5 \times 8836 \text{ cm}^2$. $5 \times 8836 = 44180 \text{ cm}^2$.

4. Convert the surface area from square centimeters to square meters: We are given $1$ meter = $100$ centimeters. So, $1 \text{ m}^2 = (100 \text{ cm})^2 = 10000 \text{ cm}^2$. To convert from $\text{cm}^2$ to $\text{m}^2$, divide by $10000$. Surface area in $\text{m}^2$ = $\dfrac{44180 \text{ cm}^2}{10000 \text{ cm}^2/\text{m}^2}$. Surface area in $\text{m}^2$ = $4.4180 \text{ m}^2$. Surface area in $\text{m}^2$ = $4.418 \text{ m}^2$.

Conclusion:The exterior surface area of the box without a lid is $4.418$ square meters.