No, you do not need insurance coverage since you do not own a vehicle unless you are in a situation where the courts (or state) require you to carry financial responsibility coverage, but you don't own a vehicle.
If you drive someone else's vehicle, their insurance will cover you as long as you have permission, unless a specific exclusion applies. If you are living at home with your parents or with someone else and will be driving their car they should contact their insurance company to determine if you are covered.
You may also consider purchasing a "nonowners" insurance policy. Many insurance companies offer a nonowners policy for people who drive occasionally but don't own their own car. Nonowners policies typically include liability, medical payments, and uninsured/underinsured motorist coverages. Nonowners policies do not include coverage for a vehicle like comprehensive, collision, towing reimbursement, or rental reimbursement coverage.
What's more, a nonowners policy provides coverage for any car you drive, not just rental cars. If you borrow a car from a friend and get into an accident in a borrowed car, your friend's insurance would kick in first. But if, for example, the accident was your fault and the damage to the other driver's property exceeded the liability limits on your friend's policy, your nonowners insurance policy would cover the excess (up to policy limits). A nonowners would not cover to fix your friend's car.
You can shop for insurance by getting a nonowners policy quote here or if you already own a car, get a car insurance quote.