Here the base field is the field of complex numbers.

Any $K3$ surface $S$ of degree $8$ in $\mathbb{P}^5$ is contained in $3$ linearly independent quadrics. It can be seen that in the general case $S$ is a complete intersection of $3$ quadrics. However, there are some special cases where this is not true, but they can be completely described. The point is that $S$ is a complete intersection of $3$ quadrics if and only if a general hyperplane section (which is a smooth curve of genus $5$) has no $g_3^1$. You can find all details in [Beauville, Complex Algebraic Surfaces, Chapter VIII, Example 14 and Exercise 11]. 

