I have recently been reading about the unpatchable BadUSB and similar malware. I use Chrome OS, which is supposed to be very secure, but could someone with physical access to my system compromise it with malware such as this?
TL;DR: ChromeOS is vulnerable to BadUSB, but this is probably not a big security concern.
BadUSB works by installing malicious code in a USB device controller. This can allow one type of device to impersonate another. For example, a thumb drive may impersonate a keyboard; one that starts typing commands right when it is plugged in. While operating systems have moved away from autorun functionality to prevent an attack from happening the moment you plug in a drive, OSs still allow a keyboard to begin typing without any security prompts. So any OS that allows a keyboard (or mouse or potentially other USB devices) to immediately start interacting with the computer is potentially vulnerable to BadUSB. Unless I'm mistaken, this is true for Chrome OS.
That said, I doubt that this should be your biggest security concern. BadUSB is difficult to execute. It involves finding a hackable USB device (it seems this isn't difficult as USB microcontrollers aren't well protected), creating software appropriate to the device type and the attack you are planning, hacking the device (either physically or by flashing the microcontroller), engineering the process of getting the device used on the target computer, and then being able to perform the attack without being noticed. Note that this last part is non-trivial as this is not some backdoor, over-the-net type attack. This is a USB device that unexpectedly starts typing characters, albeit very quickly, while you are looking at your computer.