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Non-Newtonian viscoelastic fluid flow past an isothermal sphere embedded in non-Darcy porous medium is examined numerically in this work. To be specific, the non-Newtonian Powell–Eyring fluid in the presence of both the heat and mass characteristics is mathematically modelled in terms of differential system. A non-Darcy drag force model is employed to simulate the effects of linear porous media drag and second-order Forchheimer drag. The surface of the sphere is maintained at a constant temperature and concentration. The numerical solution of the resultant system is reported via the Keller box method. Both tabular and graphical forms are adopted to identify the variations in Powell–Eyring fluid velocity, Powell–Eyring fluid temperature, and Powell–Eyring fluid concentration. In addition, the surface physical quantities, namely, skin friction and heat and mass transfer rates, are explored. The obtained observations are validated with earlier Newtonian studies. We found an excellent match in this regard. It is found that the velocity is reduced with increasing fluid parameter (ε), Forchheimer parameter (Λ), and tangential coordinate (ξ). In contrast, the temperature and concentration are increasing with increasing value of ε. A very slight increase in velocity is seen with an increase in the local non-Newtonian parameter, δ. But the temperature and concentration decrease slightly with an increase in δ. An increase in Darcy parameter enhances velocity but reduces both temperature and concentration. The present study finds an extensive array of applications in modern nuclear engineering, mineral and chemical process engineering, nuclear waste in geomaterial repositories, petroleum product filtration, and insulation systems.