Indeed this space is never compact except for trivial cases (i.e. the compact set $X$ having only finitely many points). For more information on the topology (which is colloquially known as the compact-open topology) of the space you are asking about, see e.g. the book Engelking: General topology. (Chapters 2.6 and 3.4 will have most of the information you could possibly need).

Indeed this space is (almost, see the comment by YCor) never compact except for trivial cases (i.e. the compact set $X$ having only finitely many points). For more information on the topology (which is colloquially known as the compact-open topology) of the space you are asking about, see e.g. the book Engelking: General topology. (Chapters 2.6 and 3.4 will have most of the information you could possibly need).

Thanks for the clarification by @YCor, the never compact of the original answer was a bit hasty and comes from my use of these spaces where $X$ is always locally euclidean (i.e. the $X$ should be a manifold). Which is of course not necessary.