Install pacman on Fedora (dirty, right?):
fedora ~]# dnf install arch-install-scripts
fedora ~]# pacman-key --init
fedora ~]# pacman-key --populate archlinux
Enable Arch mirrors from:
fedora ~]# /etc/pacman.d/mirrorlist
Create the new directory:
fedora ~]# mkdir /path/to/Arch
Install Arch in said directory:
fedora ~]# pacstrap -di /path/to/Arch base base-devel sudo
If it spews errors about missing dependencies, just ignore them:
:: Do you wanto to skip the above packages for this upgrade? [y/N] y
SELinux will go haywire so you should create a couple local policies and change a couple contexts, or set it to permissive mode if you’re lazy
When that’s sorted that’s it and Arch is installed, ready to be chrooted into.
What? Were you surprised it was this quick? 
systemd-nspawn is pretty neat, but SELinux thinks otherwise so set it to permissive when launching it
fedora ~]# setenforce 0
fedora ~]$ sudo systemd-nspawn -bD /path/to/Arch/
If you feel uncomfy, re-enable SELinux
fedora ~]# setenforce 1
Enjoy your Arch.
To forward X from the chroot environment to the host, you need to allow local network access to your host’s X server
fedora ~]$ xhost +local:
fedora ~]$ echo $DISPLAY
:0.0
And throw the chroot environments X to the host, which identified in this case as :0.0
arch ~]$ export DISPLAY=:0.0
Sound can be enabled by installing ‘paprefs’, and ticking the box that says ‘Enable network access to local sound devices’
EDIT: Oh, and you probably don’t want to be in an untrusted network when forwarding X & pulseaudio.