From what I've read it seems to be the answer to everything that ails docker. Images in user directory, containers with only user permissions, no daemon, etc. Yet every guide to installing seems to rely on the system's package manager, and the build seems to rely on packages being installed (runc, common), that would also require permissions to the system's package manager.
So how can I get podman to run on a system where I only have normal user access?
If the required dependencies are already installed, you could try to build it from source.
Though, if prerequisites like enabled user namespaces are not met, you are out of luck, unfortunately.