The answer depends on your situation. You will not need a car seat if you don't ever plan on riding in a car with your baby. This is possible in places with excellent public transport networks or for families who live right in the center of town, where they have everything they need. But even if this describes your situation, you may still want to keep in mind that you may like to have the option of taking a taxi to the doctor's office or anywhere else. In many places, car seats are not compulsory in cabs.
But that doesn't mean that your baby will be safe without one, so if you ever want to take a cab anywhere, having a car seat on hand may be good. Car seats can be bought inexpensively at places like Target or Walmart, and while they do take up space, car seats can also be convenient for around the house. We used our car seats to rock the baby to sleep or so that baby would not freak out while I was taking a quick shower the car seat was convenient to place in the bathroom!
For families who want to ride in relatives' or friends' cars occasionally, there is no question that they do need a seat. If grandma is planning on driving the baby in her car, for instance, you may have the option of keeping the seat installed in her car instead of having it in your house. Do you need a car seat if you don't have a car? Surprisingly, in many cases, the answer is yes. You might also like to read about surviving car journeys with an infant.