The plane is divided into unit cells, and each of the cells is painted in one of two given colors. Find the minimum possible number of cells in a figure consisting of entire cells which contains each of the $16$ possible colored $2\times2$ squares.
1. **Identify the problem**: We need to find the minimum number of cells in a figure that contains all possible $2 \times 2$ colored squares, given that each cell is painted in one of two colors.

2. **Understand the $2 \times 2$ squares**: There are $2^4 = 16$ possible $2 \times 2$ squares since each of the 4 cells in a $2 \times 2$ square can be one of two colors.

3. **Consider the arrangement of these squares**: We need to arrange these $16$ different $2 \times 2$ squares in a larger grid such that each $2 \times 2$ square appears at least once.

4. **Define the grid dimensions**: Let the grid have $m$ rows and $n$ columns. Each $2 \times 2$ square can be identified by its top-left cell. Therefore, the grid must have at least $m-1$ rows and $n-1$ columns to accommodate all $2 \times 2$ squares.

5. **Calculate the minimum area**: To ensure that all $16$ $2 \times 2$ squares are present, we need at least $m \times n \geq 16$. The smallest possible values for $m$ and $n$ that satisfy this are $m = 4$ and $n = 4$, giving $m \times n = 16$.

6. **Consider the boundary conditions**: To ensure that all $2 \times 2$ squares are included, we need to consider the boundary cells. Each $2 \times 2$ square overlaps with adjacent squares, so we need to add extra rows and columns to cover all possible configurations.

7. **Calculate the total number of cells**: The minimum grid that can contain all $16$ $2 \times 2$ squares without missing any configuration is a $5 \times 5$ grid. This is because a $4 \times 4$ grid would only cover $3 \times 3$ $2 \times 2$ squares, missing some configurations.

8. **Verify the solution**: In a $5 \times 5$ grid, there are $4 \times 4 = 16$ possible $2 \times 2$ squares, which is sufficient to cover all configurations.

Therefore, the minimum number of cells required is $5 \times 5 = 25$.

The final answer is $\boxed{25}$.