My NPO's website is on a proprietary CMS that is totally managed from a browser. There is no access to cPanel or FTP. The SLA does not limit storage space or bandwidth - at least we are not approaching any limits. Is there a way to roughly determine or estimate size, RAM and storage space of the site?
If you don't have shell access to the server, a control panel reporting this to you, or the ability to add your own web files (ilike a PHP or Perl script), then you will not be able to get this information. And even then, it may be a moot point as it sounds like you are using a SaaS platform. This means the CMS software is being universally offered to all of the host's clients. You are just taking up space in their database and maybe in a directory if they store and binary files like images there. You aren't taking up physical space like a traditional website does with its server side files, images, etc.
If you are wanting to get a rough idea of the size of the site in order to transfer the contents into an alternative CMS on another host then you could perhaps try downloading the site with a website copier (eg. HTTrack) - if it is spiderable - in order to get a quick idea of the size of all the resources (content, images, videos, CSS, client-side scripts, etc.) and then add on the size and documented requirements of your new CMS.
I am not aware of your CMS features, but if your CMS uses a database and you have direct access to that database, study the schema briefly and do a query such as
SHOW TABLE STATUS and look at the data length of the table(s) that contain content and meta-data. Take the total plus any resources within the web space.
This is what I do. Otherwise, I am not sure.