Tony, I am afraid that it's a very dirty word... basically it's a dialectal form for the word indicating the male genital organ. So "sitting at home doing gotz" would actually be (in italian): "seduto a casa facendo un c...o" . In english the closest correspondence would be "doing a f..k" (not the same exact meaning, but the same verbal strength).
I heard this expression in a few episodes of The Sopranos and was both amused and shocked when I saw that Tony Soprano had named his boat Stugotz.