Can you make a website with just a domain name?

Domain names by themselves do not allow you to run a website. All websites are ran on servers (a host) and such a server would need internet access if it's to be publicly accessed (including you if it's not on a machine within your vicinity). Some web host providers may have a deal where you seemingly just pay for the domain name, however the price would be higher (to pay for the actual hosting service and profit) and you should be careful with dealers that may restrict your domain name or hold it for ransom. Ideally, one should register (and pay for) their domain name through a reputable registrar themselves and opt for a deal with a host provider, whether an easy-to-use web host, Virtual Private Server or dedicated server - then configure one's DNS to point to those hosts. Otherwise one can run their own webserver, but consider the drawbacks, security and reliability issues especially if it's behind a domestic internet connection.

In short, it needs a server (computer) to run the webserver and website, yet it's possible to run a website without a domain name. Remember having a domain name doesn't mean you need to set up a website either.