 Solution

Take a reduced fraction $a/b\in R$. If $a\ne 0$, its only possible inverse in $\mathbb{Q}$ is $b/a$. Since $\gcd (a,b)=1$, this is in lowest terms up to sign and belongs to $R$ exactly when $\gcd (a,30)=1$. Zero is not invertible. ThereforeEquivalently, these are the rational numbers whose valuations at $2,3,5$ are all zero.