Spree currently is not a SaaS (Software-as-a-Service) aplication, you need to self-host it. However there are no limitations to what hosting provider or type you can choose from. You can run Spree in the cloud (AWS, Microsoft Azure, Google Cloud, etc), VPS (DigitalOcean, etc) or dedicated servers.
Spree is a Ruby on Rails application and all resources / guides that cover Ruby on Rails are applicable to Spree as well.
In this section we’ve provided some resources and guidelines for popular services to help you with your Spree hosting and deployment issues.