Strain-free GaAs cone–shell quantum structures (CSQS) with widely tunable wave functions (WF) are fabricated using local droplet etching (LDE) during molecular beam epitaxy (MBE). During MBE, Al droplets are deposited on an AlGaAs surface, which then drill low-density (about 1 × 107 cm−2) nanoholes with adjustable shape and size. Subsequently, the holes are filled with GaAs to form CSQS, where the size can be adjusted by the amount of GaAs deposited for hole filling. An electric field is applied in growth direction to tune the WF in a CSQS. The resulting highly asymmetric exciton Stark shift is measured using micro-photoluminescence. Here, the unique shape of the CSQS allows a large charge–carrier separation and, thus, a strong Stark shift of up to more than 16 meV at a moderate field of 65 kV/cm. This corresponds to a very large polarizability of 8.6 × 10−6 eVkV −2 cm2. In combination with simulations of the exciton energy, the Stark shift data allow the determination of the CSQS size and shape. Simulations of the exciton–recombination lifetime predict an elongation up to factor of 69 for the present CSQSs, tunable by the electric field. In addition, the simulations indicate the field-induced transformation of the hole WF from a disk into a quantum ring with a tunable radius from about 10 nm up to 22.5 nm.