Steam keeps logging me out of the mobile application
Solution 1:
More of a superuser or apple.stackexchange question, but this is most likely because of behavior of iOS itself and not the app. iOS automatically clears the cache on the least-used apps when the device is "low on space". I don't know how the iOS Steam app is setup, but maybe it's not separating the data properly -- have you tried submitting a bug report somehow, or talking to Steam Support?
The thread I linked to mentions estimations of the amount of free space for iOS 7, but it's up to 11 now so those numbers have likely changed.