This question came up in a recent research we hold with Lubotzky, Sauer and Weinberger. We came across this post while looking online for its solution. I will share our findings.
O (Original question): Which compact locally symmetric spaces are odd dimensional rational homology spheres?
PO (Partial answer to O): For a semisimple Lie group $G$, a maximal compact subgroup $K<G$ and a cocompact lattice $\Gamma<G$, $\Gamma\backslash G/K$ is an odd dimensional rational homology sphere only if $G$ is locally isomorphic to $\text{SO}(n,1)$, with the single possible exception $G=\text{SL}_3(\mathbb{R})$.
Erasing the word "locally" in O, we get the following.
R (Related question): Which compact symmetric spaces are odd dimensional rational homology spheres?
For this we actually have a full solution.
FR (Full answer to R): A compact group $U$ and a closed subgroup $K<U$ give rise to a symmetric space $U/K$ which is an odd dimensional rational homology sphere iff (up to center) either $U=\text{SU}(3)$ and $K=\text{SO}(3)$ or $U=\text{SO}(n+1)$ and $K=\text{SO}(n)$ ($n$ even).
Note that the only case which is not an actual a sphere is $\text{SU}(3)/\text{SO}(3)$, which is the dual symmetric space of $G=\text{SL}_3(\mathbb{R})$.
In this case, $\Gamma\backslash G/K$ is not an odd dimensional rational homology sphere at least for some cocompact lattice $\Gamma<G$ by the beautiful discussion here.
Proof of FR:
[1, Theorem 1] gives that (up to center) either $U=\text{SU}(3)$ and $K=\text{SO}(3)$ or $U/K$ is an actual odd dimensional sphere.
In [2]* there is a classification of all presentations of odd dimensional spheres as a homogeneous spaces, that is a compact Lie groups mod a closed subgroup. Comparison with the table of compact symmetric spaces and inspection reveals that the only pairs in the classification which give rise to symmetric spaces are $\text{SO}(n)<\text{SO}(n+1)$ for $n$ even.
$*$ I couldn't find the actual reference [2], but the result is described in the MatchSciNet review, as well as in [3, p. 179].
Proof of FR $\Rightarrow$ PO:
Let $U/K$ be dual symmetric space of $G/K$.
Denoting by $H^n_{\text{cg}}$ the continuous group cohomology and by unlabeled $H^n$ the space cohomology, we have $H^n_{\text{cg}}(G,\mathbb{C}) \cong H^n(U/K,\mathbb{C})$ and Shapiro Lemma gives
$$H^n(\Gamma\backslash G/K,\mathbb{C})\cong H^n_{\text{cg}}(\Gamma,\mathbb{C})\cong H^n_{\text{cg}}(G,L^2(G/\Gamma)) \cong H^n_{\text{cg}}(G,\mathbb{C})\oplus H^n_{\text{cg}}(G,L^2_0(G/\Gamma)).$$
Therefore we get an injection $H^n(U/K,\mathbb{C}) \hookrightarrow H^n(\Gamma\backslash G/K,\mathbb{C})$.
We thus get a necessary condition: If $\Gamma\backslash G/K$ is a rational cohomology sphere than so is the dual symmetric space of $G$ (note that this condition does not depends on $\Gamma$).
We are done by using FR and noting that $\text{SU}(3)/\text{SO}(3)$ is the compact dual of $\text{SL}_3(\mathbb{R})/\text{SO}(3)$ and $\text{SO}(n+1)/\text{SO}(n)$ is the compact dual of $\text{SO}(n,1)/\text{SO}(n)$.
[1] Wolf, Joseph,
Symmetric spaces which are real cohomology spheres.
J. Differential Geometry 3 (1969), 59–68.
[2] Borel, Armand
Le plan projectif des octaves et les sphères comme espaces homogènes. (French)
C. R. Acad. Sci. Paris 230 (1950), 1378–1380.
[3] Besse, Arthur L.
Einstein manifolds.
Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 10. Springer-Verlag, Berlin, 1987.
