Cats were known and domesticated in ancient Egypt, Greece, and Rome, but are absent from the Bible and Second Temple literature. The Persians despised cats, but the Talmud tolerates them.Lots of interesting information here, especially about the Talmudic period.
One detail: cats do appear once in Second Temple literature. Epistle of Jeremiah 22 describes cats perching on the idols in pagan temples. These are presumably domesticated cats if they are hanging around in temples.
That shows that Second Temple Jews knew of cat domestication, but not necessarily that they kept cats themselves. (A Greek manuscript of the Epistle of Jeremiah was found among the Dead Sea Scrolls [7Q2], so its Second Temple Jewish origin is secure.)
For more on cats in antiquity and the ancient biblical world, see here and links, plus here and here.
Visit PaleoJudaica daily for the latest news on ancient Judaism and the biblical world.