MBAir to Chromebook?
You have multiple options depending on your needs.
The most straight-forward option would be to format your external disk in the ExFAT format. This format is supported for both read and writes by both macOS and Chrome OS. It also supports large files, which is otherwise a problem with older versions of FAT.
Another option would be to format the external disk with HFS+. This format can be read and written to from macOS, and you can read the files from Chrome OS. You'll need to copy them over to other storage (internal/external) once on Chrome OS, as you will be able to only read the files - not write to them again on the HFS+ formatted disk.