I must confess I don't have an exact size yet, but for now I'm going to leave it to judgment. There is also the possibility that the content's subjective "annoyingness" could play an influence if the avatar or signature is considered borderline violating the rule on "tasteful and reasonable in size".
But, to provide some idea, if your avatar is greater than 200KB, and signature around 300KB, it's probably too big. (Most are far smaller.) Generally avatars should be limited to around 150px in width or height.