I was trying my best to run containers within containers just like you for the past few days. Wasted many hours. So far most of the people advise me to do stuff like using the docker's DIND image which is not applicable for my case, as I need the main container to be Ubuntu OS, or to run some privilege command and map the daemon socket into container. (Which never ever works for me)

The solution I found was to use Nestybox on my Ubuntu 20.04 system and it works best. Its also extremely simple to execute, provided your local system is ubuntu (which they support best), as the container runtime are specifically deigned for such application. It also has the most flexible options. The free edition of Nestybox is perhaps the best method as of Nov 2022. Highly recommends you to try it without bothering all the tedious setup other people suggest. They have many pre-constructed solutions to address such specific needs with a simple command line.

The Nestybox provide special runtime environment for newly created docker container, they also provides some ubuntu/common OS images with docker and systemd in built. Their goal is to make the main container function exactly the same as a virtual machine securely. You can literally ssh into your ubuntu main container as well without the ability to access anything in the main machine. From your main container you may create all kinds of containers like a normal local system does. That systemd is very important for you to setup docker conveniently inside the container.

One simple common command to execute sysbox:

    dock run --runtime=sysbox-runc -it any_image

If you think thats what you are looking for, you can find out more at their github: https://github.com/nestybox/sysbox

Quicklink to instruction on how to deploy a simple sysbox runtime environment container: https://github.com/nestybox/sysbox/blob/master/docs/quickstart/README.md