Is it possible to make a running website without taking any credit?

Just make sure you buy the domain identity protection when you buy the domain name, else a lookup will reveal to whom the domain is registered

Yes possible, I do it myself. Though if Govt authorities want, they can surly get your identity. Also, the website is into business dealings, then you may have to divulge the details to go by legal compliance. But if it is just blog like website, yes you can do it anonymously