I was wondering if there was around a hosting as those that have existed for years for PHP that would give the opportunity to publish many Ruby and Rails applications and not as Heroku that forces a single application for dyno.
In classical hosting PHP I can create a folder, upload some files in php and navigate them through links.
You can something like that on some Web hosting?
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For small production projects, I use webfaction, it's easier to push to production than to configure a complete VPS as it's more like a managed hosting (with all the tools and documentation you need for rails) . I use mina for deployment and Git for version control.
To complete @Tadman answer you can check OpenShift if you want a more Heroku like alternative.
When I started using Rails I was also tempted to compare and seek for a 'php-hosting' like solution. But it's just a different approach.
To answer your question more precisely, you don't drop files in a folder and navigate with links in a classic rails project. You have to understand the concept of MVC, routing ...
I suggest that you give the rails-tutorial a try, it is a good starting point for understanding the whole rails ecosystem.
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You can try Ruby hosting on Jelastic PaaS with automated deployment to containers and scaling, as well as pay-per-use pricing model that makes it not so pricy.
There are pre-configured Apache and NGINX containers for running Ruby application, supported different Ruby versions, built-in Ruby on Rails framework, Passenger, Puma, Unicorn, Bundler dependency manager etc.
When deploying a Ruby application, only a single context (ROOT) can be used. However, you can switch between three deployment types: deployment, production and test.
More details are described here https://jelastic.com/blog/ruby-paas-hosting/
You can start with a free trial and test how it suits your project before investing any budget. This Ruby PaaS is available on different local service providers https://jelastic.cloud/
For small projects, Heroku is really the best deal. Their free tier does everything you'd need for something that's occasionally used and doesn't have a lot of scaling issues. You are restricted to one application per "dyno", per account.
For anything more demanding it's not hard to set up a hosting environment on a VPS provider. Although it takes some additional knowledge, you'll be able to set up and configure a web server using a tool like Passenger and manage your own instances. For $10/mo. you can have a very capable server instance that will handle way more than a $7/mo. dyno can manage. Even the $5/mo. server from a provider like Digital Ocean is a fantastic deal.
PHP's fire and forget method of hosting is convenient, but it's actually a lot more work in the long haul compared to an efficient workflow based around Rails and Passenger.
For example, using Capistrano and a version control system like Git you can make changes, test locally, package up into a commit and deploy on your server within minutes. It's basically effortless once you get it working.