The only solution is can think of, aside from carrying the data with you like you are doing right now, is obvisouly to access it online.
Given that you have 5TB of data, most regular online backup solutions that I know of (like Dropbox) simply are too small or too expensive when reaching such large storage numbers.
Since you already are backing up your data to the desktop at home, have you considered setting up a FTP server on that computer? Or maybe even using some sort of remote connection to work on the data on the desktop computer itself.
Getting 5TB of data online every time you use it is of course an overkill, but you could continue to use you external hard-drive for day-to-day purposes and each time you change the data, you would upload the changes to your home FTP server. Protecting you from dataloss in case of loss/theft/failure of the external hard drive, while at the same time avoiding endless download of data when needed.
Anyway, that's my two cents. I guess we would need further description of how you work with the data to give you a better recommendation.