Construct a CFG for $L = \{ w \in \{0,1\}^*\text{ } |\text{ } w = w^R \text{ and } |w| \text{ is even}\}$
As HendrikJan stated, we can for example obtain the following languages with above CFG
$$01, 0001,\dots$$
which do not meet the required conditions.
Instead, we need to make sure that the ends are always the same. We can do that by the following production rules.
$$
S\rightarrow 0S0$$
$$S\rightarrow 1S1$$
$$S\rightarrow \epsilon$$