A subset $S$ of the set of integers 0 - 99 is said to have property $A$ if it is impossible to fill a crossword-puzzle with 2 rows and 2 columns with numbers in $S$ (0 is written as 00, 1 as 01, and so on). Determine the maximal number of elements in the set $S$ with the property $A.$
1. **Understanding the Problem:**
   We need to find the maximum number of elements in a subset \( S \) of the integers from 0 to 99 such that no 2x2 crossword puzzle can be filled with numbers from \( S \). This means that no digit can appear both as the first digit of some number and as the last digit of some number.

2. **Rephrasing the Condition:**
   The condition can be rephrased as follows: 
   - If a digit appears as the first digit of some number in \( S \), it cannot appear as the last digit of any number in \( S \), and vice versa.

3. **Counting Digits:**
   Let \( x \) be the number of digits that appear only as the first digit of some number in \( S \).
   Let \( y \) be the number of digits that appear only as the last digit of some number in \( S \).

4. **Total Digits:**
   Since we are dealing with digits from 0 to 9, we have:
   \[
   x + y = 10
   \]

5. **Number of Elements in \( S \):**
   Each digit that appears only as the first digit can pair with any digit that appears only as the last digit to form a valid number in \( S \). Therefore, the number of possible elements in \( S \) is:
   \[
   xy
   \]

6. **Maximizing \( xy \):**
   To maximize \( xy \) given \( x + y = 10 \), we use the Arithmetic Mean-Geometric Mean Inequality (AM-GM Inequality):
   \[
   \frac{x + y}{2} \geq \sqrt{xy}
   \]
   Substituting \( x + y = 10 \):
   \[
   \frac{10}{2} \geq \sqrt{xy}
   \]
   \[
   5 \geq \sqrt{xy}
   \]
   Squaring both sides:
   \[
   25 \geq xy
   \]

7. **Achieving Equality:**
   Equality in the AM-GM Inequality occurs when \( x = y \). Therefore, setting \( x = y \):
   \[
   x + x = 10
   \]
   \[
   2x = 10
   \]
   \[
   x = 5
   \]
   Thus, \( y = 5 \) as well.

8. **Conclusion:**
   The maximum number of elements in \( S \) is:
   \[
   xy = 5 \times 5 = 25
   \]

The final answer is \(\boxed{25}\).