Basically the way sudo and doas work is that they turn your current session into a privileged one, then run the command, then put your session back the way it was, this can cause security issues. The way run0 works is that it just asks systemd to do it for you, removing those security risks.

At least thats the way I understand it, im not an expert

Someone did a ELI3 explanation for this a couple days ago. The ELI5 explanation was more complicated so someone asked for ELI3 lol

ELI3

Pouring a cup of juice is something an adult needs to be involved with.

sudo is when you ask for permission to pour your own cup of juice. You ask an adult, they give you the cup and the juice, and then you’re responsible for pouring it. If the adult isn’t paying attention they may leave the fridge open for you to go back for more juice or another beverage, but otherwise you’re limited to the amount of juice the adult has given you.

run0 is when the adult just gets you a cup of juice. You tell them what you want, they go and pour the juice, and just give you the cup with the juice in it. You never enter the kitchen, so you don’t have access to the fridge, just your cup of juice.

ELI5

Basically, the SUID bit makes a program get the permissions of the owner when executed. If you set /bin/bash as SUID, suddenly every bash shell would be a root shell, kind of. Processes on Linux have a real user ID, an effective user ID, and also a saved user ID that can be used to temporarily drop privileges and gain them back again later.

So tools like sudo and doas use this mechanism to temporarily become root, then run checks to make sure you’re allowed to use sudo, then run your command. But that process is still in your user’s session and process group, and you’re still its real user ID. If anything goes wrong between sudo being root and checking permissions, that can lead to a root shell when you weren’t supposed to, and you have a root exploit. Sudo is entirely responsible for cleaning the environment before launching the child process so that it’s safe.

Run0/systemd-run acts more like an API client. The client, running as your user, asks systemd to create a process and give you its inputs and outputs, which then creates it on your behalf on a clean process tree completely separate from your user session’s process tree and group. The client never ever gets permissions, never has to check for the permissions, it’s systemd that does over D-Bus through PolKit which are both isolated and unprivileged services. So there’s no dangerous code running anywhere to exploit to gain privileges. And it makes run0 very non-special and boring in the process, it really does practically nothing. Want to make your own in Python? You can, safely and quite easily. Any app can easily integrate sudo functionnality fairly safely, and it’ll even trigger the DE’s elevated permission prompt, which is a separate process so you can grant sudo access to an app without it being able to know about your password.

Run0 takes care of interpreting what you want to do, D-Bus passes the message around, PolKit adds its stamp of approval to it, systemd takes care of spawning of the process and only the spawning of the process. Every bit does its job in isolation from the others so it’s hard to exploit.

Sudo is a setuid binary, which means it executes with root permissions as a child of of the calling process. This technically works, but gives the untrusted process a lot of ways to mess with sudo and potentially exploit it for unauthorized access.

Run0 works by having a system service always running in the background as root. Running a command just sends a message to the already running seevice. This leaves a lot less room for exploits.

Sorry, but it's also beyond my understanding. As far as I can understand, run0 doesn't use a sandbox, but will better isolate the privileged processes from unprivileged ones when executing a command.