Find the largest natural number $ n$ for which there exist different sets $ S_1,S_2,\ldots,S_n$ such that:

$ 1^\circ$ $ |S_i\cup S_j|\leq 2004$ for each two $ 1\leq i,j\le n$ and
$ 2^\circ$ $ S_i\cup S_j\cup S_k\equal{}\{1,2,\ldots,2008\}$ for each three integers $ 1\le i<j<k\le n$.
1. **Define the universal set and complementary sets:**
   Let \( U = \{1, 2, \ldots, 2008\} \) and define \( A_k = U \setminus S_k \) for each \( k \). This means \( S_k = U \setminus A_k \).

2. **Reformulate the conditions:**
   - The first condition \( |S_i \cup S_j| \leq 2004 \) translates to \( |U \setminus (A_i \cap A_j)| \leq 2004 \), which simplifies to \( |A_i \cap A_j| \geq 4 \).
   - The second condition \( S_i \cup S_j \cup S_k = U \) translates to \( U \setminus (A_i \cap A_j \cap A_k) = U \), which simplifies to \( A_i \cap A_j \cap A_k = \varnothing \).

3. **Analyze the set \( U_0 \):**
   Let \( U_0 = A_1 \cup A_2 \cup \cdots \cup A_n \subseteq U \). Each element of \( U_0 \) corresponds to an unordered pair \(\{i, j\} \subseteq \{1, \ldots, n\}\), and each pair appears at least 4 times.

4. **Count the pairs:**
   Since each pair \(\{i, j\}\) appears at least 4 times, we have:
   \[
   |U_0| = \sum_{i < j} (\text{number of times pair } \{i, j\} \text{ occurs}) \geq 4 \binom{n}{2} = 4 \cdot \frac{n(n-1)}{2} = 2n(n-1)
   \]

5. **Bound the size of \( U_0 \):**
   Since \( U_0 \subseteq U \) and \( |U| = 2008 \), we have:
   \[
   2008 \geq |U_0| \geq 2n(n-1)
   \]

6. **Solve the inequality:**
   \[
   2008 \geq 2n(n-1) \implies 1004 \geq n(n-1)
   \]
   To find the largest \( n \), solve the quadratic inequality:
   \[
   n^2 - n - 1004 \leq 0
   \]
   The roots of the quadratic equation \( n^2 - n - 1004 = 0 \) are:
   \[
   n = \frac{1 \pm \sqrt{1 + 4 \cdot 1004}}{2} = \frac{1 \pm \sqrt{4017}}{2}
   \]
   Approximating \( \sqrt{4017} \approx 63.4 \), we get:
   \[
   n \approx \frac{1 + 63.4}{2} \approx 32.2
   \]
   Since \( n \) must be a natural number, the largest possible \( n \) is 32.

7. **Verify the construction for \( n = 32 \):**
   For \( n = 32 \), consider all pairs \(\{i, j\} \subseteq \{1, \ldots, 32\}\), each appearing four times. Assign these pairs to elements of \( U \) (different pairs to different elements). Construct the sets \( A_1, \ldots, A_{32} \) such that \( x \in A_i \) if the pair at \( x \) contains \( i \). Then, construct \( S_i = U \setminus A_i \).

The final answer is \( \boxed{32} \)