Does the Bible say anything about hair length? Are men supposed to have short hair and women have long hair?

TL;DR:

Hair length is not an inherently moral or immoral decision. Rather, with our hair length we should honor God as male and female and be culturally modest.

In 1 Corinthians 11:3–15, a dispute in the town of Corinth caused the apostle Paul to write directly to the church about culture, authority, and rebellion. In Corinth, it was common for women to show their submission to their husbands by wearing a veil. Those who did not were either pagan temple prostitutes or rebellious women. A woman who attended a Christian gathering, such as church, without a veil would be confusing at best and dishonoring her husband and others at worst. Similarly, a man wearing a veil was not culturally acceptable in Corinth. In verse 6, Paul writes that women who unveil themselves may as well shave their heads. He also communicates that it would be odd for a man to wear a veil or have long hair (v. 14). In summary, Paul urges the Corinthians to fit in with the culturally acceptable standards of male and female appearance.

from the old testament

The Old Testament does not comprehensively provide explicit commands regarding hair length for men or women, but it does address hair in various contexts. For example, men and women who took the Nazirite vow were required to let their hair grow long as a sign of their dedication to God. Hair length during this vow symbolized separation and consecration to God (Numbers 6:1-21).
In some instances, long or unkempt hair was associated with shame or mourning, such as in Isaiah 3:24, where God warns that a time will come when women’s hair will become "a bald spot" as a sign of judgment.
Ezekiel 44:20 mentions that priests should not let their hair grow long, implying a form of neatness.
Hair length often had cultural significance. For example, in 2 Samuel 14:26, Absalom's hair is described as being very long and heavy, signifying beauty and prestige.

from the new testament

In 1 Corinthians 11:3–5, Paul starts his response to the situation by clarifying biblical leadership and communicates that while both men and women can do spiritual things like prophecy, they should present their bodies differently while doing so. Men should honor their bodies in the way that is culturally appropriate, as should women.
In 1 Corinthians 11:6, Paul equates a woman dishonoring her head by leaving it uncovered to the same as shaving her head entirely. Because it is disgraceful for women to do either, Paul instructs women to keep their hair long and cover it so as to be in-step with the Corinthian culture’s language of honor and respect at that time.
Paul explains the Biblical significance behind his instruction in 1 Corinthians 11:7-11. While God made woman out of man, he made man after the likeness of Himself. It is, therefore, dishonorable for a man to cover his head because he is the “image and the glory of God” (1 Corinthians 11:7). Likewise, women should have a “symbol of authority” on their heads “because of the angels” (1 Corinthians 11:10). None of these statements from Paul are communicating that women are less important or worthwhile than men; in fact, he says “in the Lord woman is not independent of man nor man of woman” (1 Corinthians 11:11).

implications for today

Some read Paul’s letter and think hair length is his main point. The broader view is that Paul is saying women should look like women and men should look like men. God created two genders. Paul may also be counseling Christians to avoid rebelling against cultural norms just to show off our "Christian liberty" and that it matters how we present ourselves. He indicates that women should show honor to the male leadership of the church, and men and women should not mix or confuse their God-ordained roles. Today, in the West, few use veils or head coverings to show submission or gender. The heart-issue is our submission to the authority of God and how we show that to those around us by the way we present ourselves.

understand

Hair length should reflect cultural norms and gender identity.
Cultural context matters in understanding the significance of hair length and appearance.
Appearance should reflect submission to God’s authority.

reflect

How does your choice of hair length reflect your understanding of cultural norms and your identity as a man or woman?
How can you embrace your gender identity and submission to God’s authority through your appearance?
What influences your perspective on hair length?

engage

How can the principles of cultural relevance in 1 Corinthians 11 be applied to modern-day Christian life without compromising biblical values?
Why does Paul emphasize the distinction between men and women in their appearance?
How does understanding the cultural context of 1 Corinthians 11 help us better interpret the Bible’s teachings on personal appearance?