Certainly it’s possible. Most web servers are just older workstations sitting in a basement somewhere. It doesn’t take any special hardware to set up… just about anything can be turned into a server. The company I used to work for used a janky old Mac G4 no one else wanted as their primary web server for years before it finally died.
But it’s not exactly the easiest thing to do if you don’t have much experience. And it also means that when something goes wrong (WHEN, mind you… not if. Servers going down for one reason or another is a fact of life.), you’re pretty much on your own.
If you’re interested in giving it a try, here’s a pretty good how-to:
https://www.dslwebserver.com/
Personally, though, unless you actually *need* the level of control having your own server provides, you should really just save yourself the headache and get on a good shared host. Site5.com and Bluehost.com are two of the best, and they’re like $4 a month.