GoDaddy shared hosting web API/RESTful

A VPS server is a little too expensive for me but I still want to have a basic web API to call to through my website. Is this possible at all through the GoDaddy shared hosting packages? The package I'm on is running on CentOS and I'm able to SSH into it but obviously I'm unable to install anything. Is there anything like an ASP.NET or NodeJS backend I can use? I've tried running a Node instance by downloading the binaries and running it locally but obviously I can't point this towards port 80 and I'm unable to access the instance remotely with any other port. Since this GoDaddy package provides easy access to Perl modules, could I maybe make a similar RESTful API with that? I'm not quite sure how to get this working but I was hoping I could maybe get something set up so I can just do a basic /api/helloworld get request as I have done before on other less restrictive platforms.