You can use the following code (drawn from my answer to this question) to determine both your database's size and the free space on the filesystem:
NSArray *paths = NSSearchPathForDirectoriesInDomains(NSDocumentDirectory, NSUserDomainMask, YES);
NSString *documentsDirectory = [paths objectAtIndex:0];
NSString *persistentStorePath = [documentsDirectory stringByAppendingPathComponent:@"database.sqlite"];
NSError *error = nil;
NSDictionary *fileAttributes = [[NSFileManager defaultManager] attributesOfItemAtPath:persistentStorePath error:&error];
NSLog(@"Persistent store size: %@ bytes", [fileAttributes objectForKey:NSFileSize]);
NSDictionary *fileSystemAttributes = [[NSFileManager defaultManager] attributesOfFileSystemForPath:persistentStorePath error:&error];
NSLog(@"Free space on file system: %@ bytes", [fileSystemAttributes objectForKey:NSFileSystemFreeSize]);
This assumes that your database is called "database.sqlite" and is stored at the root of your application's documents directory.