Yes, it is possible and rather easy to do so.
You start it with the connect/share button on the main toolbar, then selecting start content server; it can be configured from preferences > sharing. It should be a simple task, if you need more help just ask. With this method calibre must be running because basically what you are doing is setting up a web ...
I am not sure why you would need to run this on a web server; if it is just for backup reasons and distributed access I would just push the folder in one of the common online storage solutions like Dropbox or similar.
Calibre can be run as a server itself and offer the files in its library via a web interface. You can then access the files via the web ...
Alternatively, if you have a server available, with PHP, you can install COPS (https://github.com/seblucas/cops/releases) on the server, and use rsync to regularly copy your entire Calibre directory to the server. Then you won't need to have Calibre running full time.
Again, it makes sense to protect it with a password.
You can definitely create your own eBook library by combining Calibre with other solutions to ensure your books, highlights and comments are reliably accessible online which would require a lot of overhead.
I had a similar problem when we set out to build BookFusion. BookFusion allows you to seamlessly organize and store your eBooks and docuemtns in the ...
Calibre has a builtin server that can allow you to share your ebooks over the internet.
From the user manual:
Start Content server: Starts calibre’s built-in web server. When
started, your calibre library will be accessible via a web browser
from the Internet (if you choose). You can configure how the web
server is accessed by setting preferences at
Hosting the files is the easy part. Making sure that you don't violate any copyright laws in doing so is the hard part.
To host the files, all you need to do is copy the files into a folder on your web host. Then create a web page with links to those files. If you don't know how to do this, then you should check the resources offered by your web hosting ...