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?

The following paper might be of interest: Rational Analogs of Projective Planes 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. 


The Cayley plane $\mathbb{O}P^2$ is an example. See this question: 


Here is a strategy in answer to your second question. Suppose you are given a graded vector space $H_\ast$, and you wish to realise this as $H_\ast(X;\mathbb{Q})$ for some smooth closed manifold $X$. First, you must check that the linear dual $H^\ast$ carries the structure of a finite dimensional Poincaré duality algebra. Then, assuming that $H^1=0$ (equivalently $H_1=0$) you can apply the SullivanBarge theorem, which essentially says that the necessary conditions that $H^\ast$ be the cohomology algebra of a smooth simplyconnected closed manifold, are also sufficient. You'll find a clean statement in Chapter 3 of this book, but let me try anyway. Suppose your Poincaré duality algebra $H^\ast$ has formal dimension $n$. Then it can be realised by a closed simplyconnected manifold if, and only if, one of the following holds:
This is of course proved by surgery theory (I'm not sure how it relates to the reference given in Igor's answer). 


A general (surgery based, surprise...) construction is give by Kalinin in: REALIZATION OF QUADRATIC FORMS BY SMOOTH MANIFOLDS Math USSR Sbornik 62 (1989), n. 1, p 177, theorem 2.1.1 

