What is the smallest possible area of a rectangle that can completely contain the shape formed by joining six squares of side length $8$ cm as shown below?

[asy]
size(5cm,0);
draw((0,2)--(0,3));
draw((1,1)--(1,3));
draw((2,0)--(2,3));
draw((3,0)--(3,2));
draw((4,0)--(4,1));
draw((2,0)--(4,0));
draw((1,1)--(4,1));
draw((0,2)--(3,2));
draw((0,3)--(2,3));
[/asy]

$\text{(A) }384\text{ cm}^2\qquad\text{(B) }576\text{ cm}^2\qquad\text{(C) }672\text{ cm}^2\qquad\text{(D) }768\text{ cm}^2\qquad\text{(E) }832\text{ cm}^2$
1. **Understanding the Problem:**
   We need to find the smallest possible area of a rectangle that can completely contain the shape formed by joining six squares of side length \(8\) cm each. The given shape is an L-shaped figure.

2. **Analyzing the Shape:**
   The L-shaped figure consists of six squares arranged in a specific pattern. To determine the dimensions of the smallest rectangle that can contain this shape, we need to consider the overall width and height of the L-shape.

3. **Calculating the Dimensions:**
   - Each square has a side length of \(8\) cm.
   - The L-shape can be visualized as having a height of 3 squares and a width of 4 squares.
   - Therefore, the height of the L-shape is \(3 \times 8 = 24\) cm.
   - The width of the L-shape is \(4 \times 8 = 32\) cm.

4. **Finding the Area of the Rectangle:**
   - The smallest rectangle that can contain the L-shape must have dimensions equal to the height and width of the L-shape.
   - Thus, the area of the rectangle is given by:
     \[
     \text{Area} = \text{Height} \times \text{Width} = 24 \, \text{cm} \times 32 \, \text{cm} = 768 \, \text{cm}^2
     \]

5. **Conclusion:**
   The smallest possible area of a rectangle that can completely contain the given L-shaped figure is \(768 \, \text{cm}^2\).

The final answer is \(\boxed{768 \, \text{cm}^2}\).