Is it possible for a set to contain both cluster and isolated points?

All points of the empty space are both cluster points and isolated points (and neither), however there are no isolated or cluster points in the empty space.

$\{x\in\Bbb R\,:\, x=0\lor \frac1x\in\Bbb Z\}$ has isolated points and cluster points in the usual topolgy.

Here's another example: the set $S=[0,1]\cup\{2,\pi\}$, in the usual topology on $\mathbb{R}$. It has both cluster points (all points of the interval $[0,1]$) and it also has two isolated points ($2$ and $\pi$).

As for the empty set… Whether what you're trying to say is right or wrong depends on what exactly we're trying to say about it. For concreteness, let's assume we're talking about subsets of $\mathbb{R}$ (but it can be any topological space, of course).

• One interpretation is the statement "The empty set contains all of its cluster points and all of its isolated points", or in more formal notation "$\forall x\in\mathbb{R}: x \text{ is a cluster point or an isolated point of } \varnothing \implies x\in\varnothing$" is true, in the sense of being vacuously true. But it doesn't feel like what the title of your question meant.

• Another interpretation is that probably you meant "The empty set has some cluster points and some isolated points", and this statement is false — by the definitions of cluster points and of isolated points. The empty set has no isolated points, because an isolated point of a set must be an element of the set, but the empty set has no elements. And none of the points on the real number line is a cluster point for the empty set for a similar reason: the definition requires that any neighborhood of a cluster point contain an element of the set, but the empty set doesn't have any elements.