There are five people in my family.

Literal

I [of-の] family [topic-は] 5 [counter-人] exist.

The existential いる is used for people (animate), matching 家族 ('family') as a group of animate beings. 五人 ('five people') uses the irregular people-counter 人 with the reading ごにん. Standard sentence for family size — one of the first things learners memorize.