You can't express every rational number up to $\pi^2/6$. For example, $3/4$ is impossible - the sum can't contain 1, but then it's bounded above by $\pi^2/6 - 1$, which is less than $3/4$. However this sort of argument only seems to exclude the interval $[\pi^2/6, 1)$.

And it turns out (much less obviously) that that's the only obstruction - see R. L. Graham, On Finite Sums of Reciprocals of Distinct nth powers, Pacific Journal of Mathematics, 1964. In particular Corollary 1 says a rational $p/q$ is a sum of distinct reciprocals of squares if and only if $$p/q \in [0, \pi^2/6-1) \cup [1, \pi^2/6).$$

Graham wrote that this was an unpublished result of Erdos. Graham also wrote at the end of the paper that "It is not difficult to obtain representations of specific rationals" as such sums, but he gave no hint of how he did it. The proof appears to be "elementary" in the sense of not using any heavy machinery.

You can't express every rational number up to $\pi^2/6$. For example, $3/4$ is impossible - the sum can't contain 1, but then it's bounded above by $\pi^2/6 - 1$, which is less than $3/4$. However this sort of argument only seems to exclude the interval $[\pi^2/6 - 1, 1)$.

And it turns out (much less obviously) that that's the only obstruction - see R. L. Graham, On Finite Sums of Reciprocals of Distinct nth powers, Pacific Journal of Mathematics, 1964. In particular Corollary 1 says a rational $p/q$ is a sum of distinct reciprocals of squares if and only if $$p/q \in [0, \pi^2/6-1) \cup [1, \pi^2/6).$$

Graham wrote that this was an unpublished result of Erdos. Graham also wrote at the end of the paper that "It is not difficult to obtain representations of specific rationals" as such sums, but he gave no hint of how he did it. The proof appears to be "elementary" in the sense of not using any heavy machinery.