  Under fairly general assumptions, we prove that every compact invariant set
$\mathcal I$ of the semiflow generated by the semilinear damped wave equation
u_{tt}+\alpha u_t+\beta(x)u-\Deltau = f(x,u), (t,x)\in[0,+\infty[\times\Omega,
u = 0, (t,x)\in[0,+\infty[\times\partial\Omega in $H^1_0(\Omega)\times
L^2(\Omega) has finite Hausdorff and fractal dimension. Here $\Omega$ is a
regular, possibly unbounded, domain in $\R^3$ and $f(x,u)$ is a nonlinearity of
critical growth. The nonlinearity $f(x,u)$ needs not to satisfy any
dissipativeness assumption and the invariant subset $\mathcal I$ needs not to
be an attractor. If $f(x,u)$ is dissipative and $\mathcal I$ is the global
attractor, we give an explicit bound on the Hausdorff and fractal dimension of
$\mathcal I$ in terms of the structure parameters of the equation.
