Western female tourists in Qatar often receive stares from both local men and guest laborers from developing countries. Some looks may be intense or lascivious in nature, while others are simply curious glances. To avoid things escalating, avoid direct eye contact, which is made easier with a pair of dark sunglasses, and dress modestly. Long skirts or pants and shirts with sleeves that conceal your cleavage are ideal and will help you blend in and avoid excess attention.
Being a Western woman in Qatar can be an isolating experience. Many restaurants have separate sections for women and families, so you may feel closed off when dining out. Some men may not feel comfortable making eye contact or otherwise interacting with a woman they aren't related to as well. You may have to stand in a women-only line or sit in a women's waiting room if using a local business, but try to consider it a cultural experience rather than a hindrance.
If you have the opportunity to spend time with Qatari women on your trip, it's likely to be an enlightening experience. An invitation to a local woman's home will likely involve lots of food, hours of conversation and a warm, intimate atmosphere you might not find in the West. Should you attend a local wedding or other social function, Qatari women may introduce themselves and ask you questions, as you're likely to be something of a curiosity, so don't be afraid to answer and interact.
To avoid hassle, consider wearing a wedding band in Qatar, even if you're single. If men ask, tell them you're traveling with your husband, because married women are often afforded more respect. Though many hotel bars serve alcohol, drink moderately -- public drunkenness is a crime. If traveling with a romantic partner, know that unmarried couples are forbidden to share rooms. Also, don't engage in public displays of affection, such as hugging and kissing, because these are illegal and offensive to local culture.