How do I give my instance variable a getter?

wish helps you The first one accesses the ivar through the getter method. The second directly accesses the ivar. Since it's a simple, synthesized property, there's not much difference except that the first makes an additional method call. However, if the property were atomic, or dynamic, or the getter method were complicated, there'd be a difference in that the first one would actually be atomic while the second wouldn't and the first would actually trigger any complicated logic in the getter while the second wouldn't.
In simplest terms, the compiler re-writes the first call to:

[[self name] aMethod]

like below fixes the issue It was not always possible to declare instance variables in the implementation block. It has been possible for several years, but the date of that tutorial is April 8, 2010. When it was written, the author probably wanted it to work on older versions of the compiler.
Because _database is not declared @private, it is in fact possible to access the instance variable outside of the FailedBankDatabase implementation. In Objective-C, you can treat an object pointer as a struct pointer and access its instance variables like struct fields. So you could do this:

#import "FailedBankDatabase.h"

int main(int argc, char *argv[])
{
    FailedBankDatabase *db = [FailedBankDatabase database];
    printf("instance variable: %p\n", db->_database);
    return 0;
}

#import <Foundation/Foundation.h>

@class FailedBankInfo;

@interface FailedBankDatabase : NSObject

+ (FailedBankDatabase*)database;
- (NSArray<FailedBankInfo *> *)failedBankInfos;

@end

#import "FailedBankDatabase.h"
#import "FailedBankInfo.h"
#import <sqlite3.h>

@implementation FailedBankDatabase {
    sqlite3 *_database;
}

static FailedBankDatabase *theInstance;

+ (FailedBankDatabase*)database {
    if (theInstance == nil) {
        theInstance = [[FailedBankDatabase alloc] init];
    }
    return theInstance;
}

help you fix your problem In each actionPerformed() method you create a new instance of Setter_and_Getter and set a value. At the end of these methods, the scope of the method ends, therefore the Setter_and_Getter object gets removed by the garbage collector.
So there is no way to get access to these object instances later.