Mitchell's Embedding Theorem for not-necessarily-small categories
Solution 1:
Freyd proved in his paper Concrenteness (1973) that an abelian category which admits a faithful exact functor to $\mathsf{Ab}$ is well-powered; actually also the converse. There are abelian categories which are not well-powered, see MO/93853. Another example is mentioned in the foreword of the tac reprint of Freyd's book Abelian categories (2003) with details in his paper Stable homotopy (1966): The stable homotopy category embeds fully faithfully into an abelian category (which thus is not concretizable).