Earlier quoted context omitted.
Right, but the poster is saying they don't want to run a server AT ALL.
Then what are they running their website on? This is a perfect example of the infectious nature of the word cloud, people say the word cloud as if there's no server still running it! Oh noes, I have to run php code stripe has done all the work on and provided to me! Scary!
If my inventory service breaks I might have some pissed-off customers or have to re-ship some items, the odds of getting in bigger trouble than that are low. If my payment infrastructure goes down or gets compromised, the odds of lawsuits or federal/financial-org penalties are way higher.
So sure, you still probably have to have a "server" somewhere (you could probably make a fully static site for a merchant using third-party payment links, javascript, and mailto links or something--think "email as database"--but I suspect this would be neither featureful, secure, nor reliable), but the difference in what you need to maintain/serve is huge.