Is the length of the President's term set by the Constitution?

Yes it is. According to Article 2, Section 1, the term shall be four years, and George Washington established the custom of serving only two terms. However, one president (Franklin Delano Roosevelt) was able to seek and receive a third and fourth term. But after Roosevelt died in the midst of his fourth term, this led to the passage of the 22nd Amendment, which restricts all U.S. presidents to two terms only.