Not sure why your boss doesn't want to keep a backup of the DB on hand in the case of need of future restore, but I guess he has his reasons.
The nice thing about exporting this data out to text files is that you could always import them back into databases that aren't necessarily SQL Server in the future - as well as read them with text editors in a pinch. So, I would stick with a text file export.
Delimited Files:
CSV or some other delimiter is a pretty common way to go. There isn't a file size limit on the file... only the applications you might try to use to read the file. Putting a CSV back into SQL Server isn't a problem with tools like SSIS or even just the import/export wizard.
There are some nuances that might cause you to select a delimiter over the comma. For example let's say your tables contain strings with lots of commas. If that was the case you might pick pipe delimited to make it less confusing to work with.
If you have your heart set on comma delimited but you have strings with commas inside then there is an option when exporting to "qualify fields". Double quotes are often used in this case. So, for example a full name field that contains a comma but is exported to a comma delimited file would be qualified like this... "Thomas, Joe"
Fixed Width:
This is a popular export format when your fields are all fairly small. Fixed width can become a problem however if you have any (max) fields, binary, or other types that don't write well to text. Frankly, these data types don't export to delimited files very elegantly either.
Corruption:
Neither type of export to text are any more prone to corruption than another. You just have to be careful when people start opening them up in notepad, excel, or other random text editors and then save them... that is what can cause corruption. So be sure to keep a clean file backed up that you know no-one has messed with. But that brings us back to the original question... why not just keep a backup of the database?