Based on the System Requirements the QIE database would require 1 to 50 GB of disk space.
One factor to consider with regard to the amount of disk space required is the Message Retention Policy set by your organization. By default QIE is set to save message history for 14 days. This is a user setting that can be changed in the System Administration -> System Configuration -> Message Retention Policy. If your setup has changed the default setting to a higher number of days to store messages then more disk space may be required.
If your system has run out of disk space the resolution would be to increase the amount of disk space on the server.
The one caveat to this is to consider which database QIE is using on the back end. See What database platforms are supported, When should I consider migrating off of the default H2 database, and What are the minimum system requirements knowledge base articles.
