It depends on what kind of data you’re planning on storing. This answer is under the premise that you’re storing and modifying data at run time.
Contrary to what others have suggested, I would recommend against using /usr/share for storage. From the Filesystem Hierarchy Standard:
The /usr/share hierarchy is for all
read-only architecture independent
data files.
As you’re modifying data, this goes against the read-only nature of the /usr subsystem.
A seemingly better place to store your application state data would be /var, or more specifically, /var/lib. This also comes from the Hierarchy Standard. You could create a /var/lib/myapp, or if you’re also using things like lock files or logs, you could leverage /var/lock or /var/log.
Have a deeper look at the standard as a whole (linked to above)—you might find a place that fits what you want to do even better.
Like Steve K, I would also recommend using the Preferences API for application preference data.