We report solutions for solitons of the "accessible" type in globally nonlocal nonlinear media of fractional dimension (FD), viz., for self-trapped modes in the space of effective dimension 2<D≤3 with harmonic-oscillator potential whose strength is proportional to the total power of the wave field. The solutions are categorized by a combination of radial, orbital, and azimuthal quantum numbers (n,l,m). They feature coaxial sets of vortical and necklace-shaped rings of different orders, and can be exactly written in terms of special functions that include Gegenbauer polynomials, associated Laguerre polynomials, and associated Legendre functions. The validity of these solutions is verified by direct simulations. The model can be realized in various physical settings emulated by FD spaces; in particular, it applies to excitons trapped in quantum wells.