Buy or Rent? RaspberryPi vs. VPS

For a service that needs to be available to the outside world, I'd say a VPS is often the better practical solution.

Many home ISPs don't allow hosting of E-mail (SMTP) or traffic through to 80/443 (HTTP/S), so you have to have something which does this. "Self-hosters" often explain their setups with a reverse proxy which ends up on a home server.

I say this as a big supporter of RaspPi and home devices in general; unless it's for something like Home Automation, it's easier and more reliable to just run it remotely.