Noah entered the ark when he was 600 years old, specifically in the 2nd month, the 17th day of the month:
Genesis 7:6-13: (NASB 1995)
6 Now Noah was six hundred years old when the flood of water [e]came upon the earth.
7 Then Noah and his sons, his wife, and his sons’ wives with him entered the ark because of the waters of the flood.
8 Of clean animals and animals that are not clean and birds and everything that crawls on the ground,
9 they all went into the ark to Noah by twos, male and female, as God had commanded Noah.
10 Now it came about after the seven days, that the waters of the flood came upon the earth.
11 In the six hundredth year of Noah’s life, in the second month, on the seventeenth day of the month, on that day all the fountains of the great deep burst open, and the floodgates of the sky were opened.
12 The rain fell upon the earth for forty days and forty nights.
13 On this very same day Noah, Shem, Ham, and Japheth, the sons of Noah, and Noah’s wife and the three wives of his sons with them, entered the ark,
They left the ark the following year, specifically in the 2nd month on the 27th day:
Genesis 8:14-15 (NASB 1995)
14 In the second month, on the twenty-seventh day of the month, the earth was dry.
15 Then God spoke to Noah, saying,
16 “Go out of the ark, you and your wife and your sons and your sons’ wives with you.
17 Bring out with you every living thing of all flesh that is with you, birds and animals and every creeping thing that creeps on the earth, that they may [a]breed abundantly on the earth, and be fruitful and multiply on the earth.”
18 So Noah went out, and his sons and his wife and his sons’ wives with him.
19 Every beast, every creeping thing, and every bird, everything that moves on the earth, went out [b]by their families from the ark.
The Jewish calendar was based on the lunar cycle of 360 days, so Noah was on the ark a little over a year, around 370 days.