The use of the home directory is merely a recommendation, and is not a strict requirement, only for convenience (and convention). You can set it up anywhere that you have read and write access.
See also Workspace on USB or Package on USB
This should be perfectly possible, provided that your medium is
formatted with a compatible file system, as permissions (on fi dynamic
reconfigure and other executable files) are important in a Catkin
File systems like FAT, FAT32 and ExFat are not suitable for this.
Something like Ext3 or Ext4 or some other FS that supports regular
Linux file permissions should work fine.
If you use a "Windows" formatted drive, then your "Ubuntu" permissions may probably be lost, as drives formatted for Windows can not support that information. Execute and ownership permissions issues would/could arise.
It would all depend on whether you need portability in order to be able to access the workspace from a Windows machine, or if you consistently use [different] Ubuntu/Linux machines.
If you really want use an ExFAT formatted drive, then I'm not entirely au fait with the situation, as I do not use Windows at all. You probably could/would run into issues. A solution might be to delete the
build directories and re-run
catkin_make before accessing it on the new machine1.
The use of an external drive is a good idea, portability wise. You may need to reconfigure paths when you move to another host machine though.
Another option is to have a separate user for ROS, whose entire home directory is on the external disk. You would log in as that user to have your environment ready to go. This is the method that I use. However, this requires admin access to set up your login to point to the externally mounted disk containing the home directory, prior to your logging in as the ROS user. You may not have that ability, or access.
1 See How to migrate a catkin workspace? and cannot source workspace.