Mitchell's Embedding Theorem states that if $\mathcal{A}$ is a small abelian category, then there is a ring $R$ and a fully-faithful exact functor $F:\mathcal{A}\rightarrow R\mathsf{Mod}$.
To what extent does this still hold if $\mathcal{A}$ is not-necessarily-small? Does it still hold in general? Do we need to impose extra hypotheses?
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).