Is there a smooth compact manifold with rational homology vanishing except
in dimensions $0, 8, 16$ where it is $Q$? What would be a good strategy to find such a manifold?



manifold with given rational homology
The following paper might be of interest: [Rational Analogs of Projective Planes](http://arxiv.org/PS_cache/arxiv/pdf/1010/1010.3274v1.pdf) by Zhixu Su. 


It discusses the following question: For which $n$ is there a closed $2n$-manifold $M$ with rational cohomology $H^\*(M; \mathbb{Q}) \cong \mathbb{Q}[x]/x^3$ with $x$ of degree $n$? Observe that the multiplicative structure is essentially implied by the additive one by Poincare duality. 


There are, of course, the classical examples $\mathbb{CP}^2$, $\mathbb{HP}^2$ and $\mathbb{OP}^2$, but are there more $n$ possible? It is clear that $n$ has to be even for this, but actually $n$ has to be divisible by $4$. Beyond the classical examples, the next example occurs in dimension $32$, where there are already infinitely many (up to homeomorphism). This is, of course, a purely rational result - by the Hopf invariant 1 problem the examples above are the only examples for the integral analogue. 


The methods are those sketched in Mark Grant's answer. 


