NVarchar used to have a limit of 4000 characters, so that is probably what that is about.
Varchar limit was 8000 - so assuming that all export is restricted to 4000 characters any varchar column defined with 4001 - 8000 characters is going to be a problem.
There was also a (legacy) datatype TEXT and NTEXT which held unlimited length data - so if the APP used that those column(s) will be a problem too. SQL 2008 also had varchar(MAX) and NVarchar(MAX) which store unlimited text and could also be a problem. My guess is that if the APP is old that you are unlikely to have the newer varchar(MAX) but you might have the older, deprecated, TEXT datatype columns.
Presumably you legitimately have access to the SQL database? Why not just use that (instead of Access of Excel)?
If you cannot access SQL on the machine the database is on you could install SQL on another machine and (preferably) restore a Backup file from the old database to create a new database (on the new machine) or, failing that, copy the database files across (there are some special steps that you would need to follow in order to safely copy the database file, itself, hence why a Backup File would be preferable).
What were you planning to do with the data once you exported it? Perhaps you could do that direct onto a SQL database instead?