I remember when Apple released the new edition 12" MacBook back in 2015 (now discontinued). It was seemingly designed to be a small, lightweight portable workstation.
It didn't last long because like your MBA it had little storage, I think 125Gb for the base model. The idea seemed to be that this device only have the native applications plus any third party apps required for specific tasks or functions and all of the data they produced be stored on external drives.
This meant a lot of management and with its smaller 12" screen the device became unpopular and I believe it was discontinued around 2017.
So for example you could have MS Word for producing and editing documents but the default storage when you saved anything would be on an external HD. The same would apply to Photo Libraries, Movies, graphics, saved email attachments, downloads, Music, etc.
Since MacOSX 10 Apple introduced the option to store your Documents and Desktop Folders in iCloud Drive, this could be part of a solution for you but personally I dislike the idea.
For starters it requires that you have internet access at all times and that you have enough iCloud storage to contain your data.
This too only takes care of the Documents and Desktop Folders. See
Use iCloud Drive to store documents on your Mac, iPhone and iPad
Your work may not like the idea of you storing their data on a remote source.
These days you could get a small portable SSD of 2TB to store all your data and more, that would require a backup for safety but that would be my preferred option.
As for running the device from an external device, yes, that is possible but we would need to know the MBA model, number and type of ports and the macOS it's running.