Golani was born in Tel Aviv. Her father Yakov Gulnik was from Warsaw, and had lost his family in the Holocaust. Her mother Liza Goldstein was from Polish Galicia. Golani took up the violin at age 7, while her sister Bela learned the cello. Golani was a mathematics prodigy.
Golani became a pupil of the violinist Alexander Moskowsky at age 18. She switched to viola at age 21 and studied with the violist Oedoen Partos, and became a member of the Israel Philharmonic Orchestra. She married the Hungarian luthier Otto Erdesz, and in 1974 the couple moved to Canada, where she gave birth to her son Micha Eden Erdesz. She became a citizen of Canada in 1983. Erdesz manufactured for her what has been her preferred instrument throughout her career. After their divorce, Golani married German conductor Thomas Sanderling and moved to London. She later returned to Canada briefly before she married again. She currently resides in London with her husband Jeremy Joseph Fox. Golani is also a painter and has exhibited in America, Canada, Israel and the UK.