By Kunen's inconsistency in $\mathsf{ZFC}$ there is no nontrivial elementary embedding $j\colon V\prec V$.
Is it possible to have an inner model $M$ (a transitive class containing all ordinals of $V$) of $\mathsf{ZFC}$ with a nontrivial elementary embedding $M\prec M$? In particular can there be a nontrivial elementary embedding $L\prec L$? Can such an embedding be definable? What are good sources to read about this kind of questions?
Sure. That is the idea of sharps.
Specifically, $0^\#$ is equivalent to (and sometimes defined as) the statement that there is $j\colon L\prec L$. By Kunen's theorem it is impossible that $0^\#$ is in $L$ itself, but that's okay. We don't mind.
If there is a measurable cardinal, $0^\#$ exists. And $0^\#$ cannot be produced by set forcing.
Jech has a lot about it, Kanamori too. Schindler's book might contain condensed information on the topic as well.