We consider scaling limits of random quadrangulations obtained by applying the Cori-Vauquelin-Schaeffer bijection to Bienaymé-Galton-Watson trees with stably-decaying offspring tails with an exponent $α$ in (1, 2). We show that these quadrangulations admit subsequential scaling limits wich all have Hausdorff dimension $\frac{2α}{α-1}$ almost surely. We conjecture that the limits are unique and spherical, and we introduce a candidate for the limit that we call the $α$-stable sphere. In addition, we conduct a detailed study of volume fluctuations around typical points in the limiting maps, and show that the fluctuations share similar characteristics with those of stable trees.