Let's analyze the question:

The problem is to determine which among the four options represents an irrational number.

Explanation:

1. Option (1): $\sqrt{2} \times \sqrt{2}$

   • Simplify: $\sqrt{2} \times \sqrt{2} = \sqrt{2^2} = 2$.
   • This is a rational number.

2. Option (2): $\sqrt{21} \times \sqrt{\frac{3}{7}}$

   • Simplify: $\sqrt{21} \times \sqrt{\frac{3}{7}} = \sqrt{21 \cdot \frac{3}{7}} = \sqrt{9} = 3$.
   • This is a rational number.

3. Option (3): $\sqrt{2 \times 3 \times 5 \times 7}$

   • Simplify: $\sqrt{2 \times 3 \times 5 \times 7} = \sqrt{210}$.
   • $\sqrt{210}$ cannot be simplified further to a rational number. Hence, it is irrational.

4. Option (4): $\sqrt{\frac{2008}{1004} \times 1}$

   • Simplify: $\sqrt{\frac{2008}{1004}} = \sqrt{2}$.
   • This is irrational, but let's confirm further.

Final Check:

Both options (3) and (4) suggest irrational numbers because $\sqrt{210}$ and $\sqrt{2}$ cannot be expressed as exact fractions. If the question is to pick one, option (3) fits a general property test.