To be short, yes! Not a website perhaps but you will create trouble for yourself if you do not buy a domain name. If the name is in use you can try variations. Magazines for instance often use magazine behind their brand name. There are many new extensions as well that might help you out. Facebook and other social media are not a good alternative. You do not own these pages like you would your own domain. Emails with your name instead of gmail behind the . makes growing your business easier as well.
Do I need a website domain? I have a great name but it is used already.