Searching for high-entropy alloys with functional properties that emerge from their multi-scale structure, we have investigated the (GaNi)xCoCrFe (x = 0.4–1.6) system. We have characterized structure, microstructure, nanostructure and chemical composition of the individual phases in the multi-phase alloys and determined their magnetic, magnetostrictive and electrical properties. We found that the alloys are ferromagnetic and exhibit functional combination of magnetic softness and vanishing magnetostriction, classifying them as energy-efficient “supersilent” materials (inaudible to a human ear) for alternating-current (AC) electromagnetic applications in the audio-frequency range. The alloys develop a two-phase structure, a face-centered cubic (fcc) and a body-centered cubic (bcc), where the fcc phase fraction decreases, while the bcc fraction increases with the increasing (GaNi)x content. Ferromagnetism of the alloys originates from the highly nanostructured bcc phase, with the ferromagnetic Curie temperatures in the range TC = 750–700 K, depending on x. The fcc phase is not nanostructured and is paramagnetic at room temperature, but undergoes a spin glass transition at Tf≈6.4 K. The magnetic softness and vanishing magnetostriction of the alloys are both nanomagnetic phenomena. The magnetic-softness and magnetostriction parameters of the x = 1.3 and 1.6 alloys make them relevant for supersilent AC applications at low frequencies.