Until we resolve this, select each stereo track by clicking just above MUTE and Export Selected WAV (Microsoft) 16-bit so you can put a high quality show back together later if nothing else works. That’s recommended as show archive anyway.
Are you trying to save to DropBox?
It may not be complaining about the hard drive as much as physical memory. When was the last time you emptied the trash? What are W, P and X? I have had Safari connections use up all of my processor preventing other programs from working. Is your desktop full of icons and works in progress? That can cause problems.
Do you have a video open in iTunes? More memory. OS-X is pretty good about allocating memory and resources, but it’s not open ended.
It’s unknown why you can’t save, but make sure the exported WAV files play correctly without gaps or other issues. If they do, then force quit of Audacity should recover the project as long as you don’t reboot the computer.
But is there anything in the project such as envelope points that is not in the exported WAV files? You can make a new project from the WAV files by importing them into a new empty project.
Your image shows you are exporting to your Desktop, not to a USB drive. What exactly do you type when you save the file name for the project? Can you just save it called as project.aup or similar then rename it in Finder?
Does Help > Show Log… top right of Audacity show errors when you try to save? If so, what do the errors say exactly?