Let $X$ be a noetherian scheme which is not affine, such that all its irreducible components are affine schemes. What is an example of such a scheme.
If one removes noetherian hypothesis then one can simply take infinite disjoint union of affine schemes. However in this case one will have finitely many irreducible components.
In vague terms, I think one can try to glue two copies of an non integral affine scheme.
Such a scheme $X$ does not exist. One can show that if $X$ is a noetherian scheme, then $X$ is affine if and only if $X_{red}$ is affine. And if $X$ is reduced and noetherian, then $X$ is affine if and only if all of its irreducible components are affine.
You can find this in Exercise 3.3.1, 3.3.2 of Hartshorne, both of which are applications of Serre's criterion for Affineness.