Is there a simple argument (or a counterexample) to show that a holomorphically convex subset of an affine algebraic variety is a subvariety which is a compact Riemann surface?

Perhaps I should turn my comment into an "answer". Affine algebraic varieties over $\mathbb{C}$ are Stein spaces. That is, they are already holomorphically convex, and points can be separated by global holomorphic functions. The latter property implies that affine varieties can never contain compact Riemann surfaces. 

