What is the greatest number of balls of radius $1/2$ that can be placed within a rectangular box of size $10 \times 10 \times  1 \ ?$
1. **Determine the diameter of each ball:**
   Given the radius of each ball is \( \frac{1}{2} \), the diameter \( d \) of each ball is:
   \[
   d = 2 \times \frac{1}{2} = 1
   \]

2. **Calculate the number of balls that can fit along each dimension of the box:**
   The dimensions of the rectangular box are \( 10 \times 10 \times 1 \).

   - Along the length (10 units):
     \[
     \text{Number of balls} = \frac{10}{1} = 10
     \]

   - Along the width (10 units):
     \[
     \text{Number of balls} = \frac{10}{1} = 10
     \]

   - Along the height (1 unit):
     \[
     \text{Number of balls} = \frac{1}{1} = 1
     \]

3. **Calculate the total number of balls that can fit in the box:**
   The total number of balls that can fit in the box is the product of the number of balls along each dimension:
   \[
   \text{Total number of balls} = 10 \times 10 \times 1 = 100
   \]

Conclusion:
\[
\boxed{100}
\]