To determine the simplest quadratic radical among the given options, we evaluate each expression step by step:

① For $\sqrt{2}$, this expression is already in its simplest form as a quadratic radical, so it potentially qualifies.

② For $\sqrt{\frac{1}{3}}$, we can simplify this expression as follows:
$$\sqrt{\frac{1}{3}} = \frac{\sqrt{3}}{\sqrt{9}} = \frac{\sqrt{3}}{3}$$
This shows that $\sqrt{\frac{1}{3}}$ is not in its simplest form as a quadratic radical because it involves a rationalized denominator.

③ For $\sqrt{8}$, we simplify this expression as:
$$\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4}\sqrt{2} = 2\sqrt{2}$$
This indicates that $\sqrt{8}$ is not in its simplest form as a quadratic radical because it can be further simplified.

④ For $\sqrt{\frac{2}{7}}$, we simplify this expression as:
$$\sqrt{\frac{2}{7}} = \frac{\sqrt{2}}{\sqrt{7}} = \frac{\sqrt{14}}{7}$$
This shows that $\sqrt{\frac{2}{7}}$ is not in its simplest form as a quadratic radical because it involves a rationalized denominator.

Given the evaluations above, the only expression that is in its simplest form as a quadratic radical is $\sqrt{2}$, which corresponds to option ①.

Therefore, the answer is: $\boxed{A}$.