It looks like the default Arch install is not generating any of the user directories.
I’ve actually had a number of thoughts on this, and there’s been some discussion elsewhere. In the long run we’d like for there to be a “system Julia” much like there is a system Python and a system Java. This means we’d also like ther for there to be a .julia belonging to root.
Anyway, in the meantime, can you try creating $HOME/.julia? If that doesn’t work, try also creating a $HOME/.julia/environments. Failing that, if you do ]generate defaultenv this should create a directory for you called defaultenv which will contain the Project.toml and Manifest.toml that the Julia package manager can add projects to.
A little background: I’m a Manjaro user myself, but I’m pretty heavily involved with Julia and usually on the bleeding edge, so I haven’t bothered trying to use the pacman package yet. At the moment I’m still using binaries. That installation methodd should work just fine, if you want to go that route.