I don't know how to store a large number of large numbers represented as a string

I would suggest not to use string to store number. In you context strings will take more space and will be slow to operate upon.

Lets say, you want to store the number 12345678901234567890. Now if you want to store this number using string you will need 20 bytes(8 bit char is used). String s = "12345678901234567890";

Instead of that you can use array of unsigned integer. In most of the modern compiler and computer architecture integer is of 4 byte. so they can represent numbers till 4,294,967,295. Now lets consider largest number is decimal(all digits are nine) that can be represented by unsigned integer is 999,999,999. If you use array of integer now you will only need 3 integers. which is 12 bytes. unsigned int number[] = {234567890, 345678901, 12};

Now consider simple arithmetic operation addition. unsigned int number1[] = {123456789, 123456789}; unsigned int number2[] = {987654321, 987654321}; now if you want to add above two number you will only have 2 iteration to get your answer.

you can still further compress done memory foot print if you got at bit level. It would be bit tedious job but you will get best performance.

Your specific problem:

The problem you're hitting is that when you enter a number larger than a signed integer can hold, cin is put into error state. This means subsequent calls to cin don't work until you call cin.clear().

Read about it here: http://www.cplusplus.com/reference/ios/ios/clear/

To fix your code try adding cin.clear() after calls to cin like so:

#include <iostream>
#include <string>

using namespace std;

int main(int argc, const char * argv[]) {

int numOfNumbers;
int numbersAdded=0;
string input;

/* todo: use a vector<string> instead - which handles resizing and allocation for you */
string * arrayStrings;

if (!(cin >> numOfNumbers))
{
    cout << "Warning: cin failed, did you enter a number larger than an numOfNumbers can hold? Value stored: " << numOfNumbers << endl;

    /* todo: maybe stop execution here and report back? */

    /* this will reset cin and leave numOfNumbers holding the largest value it can */
    cin.clear();
}

arrayStrings = new string[numOfNumbers];
/* todo: check the allocation succeeded, otherwise stop and report the error */

while(numbersAdded<numOfNumbers){
    if (!(cin >> input))
    {
        cout << "Warning: input failed, you entered a valid string? Value stored: " << input << endl;

        /* this will reset cin and leave input holding the largest value it can */
        cin.clear();
    }

    arrayStrings[numbersAdded]=input;
    numbersAdded++;
}

for( int i = 0;i<numOfNumbers;i++){
    cout<<arrayStrings[i]<<endl;
}

/* todo: this only deletes the array of pointers, not the strings themselves. Use a vector instead to avoid having to do memory management. */
delete arrayStrings; 
return 0;
}

Running the above program will give you the following output when a large number is entered:

99999999999999999999999999999
Warning: cin failed, did you enter a number larger than an numOfNumbers can hold Value stored: 2147483647

A better approach:

As others pointed out in comments there are other things to tidy up here too, using vectors etc. Here's an alternative implementation that's cleaner and has more validation:

#include <iostream>
#include <string>
#include <vector>

using namespace std;

int main(int argc, const char * argv[]) 
{
    unsigned int numOfNumbers;
    vector<string> numberStrings;

    cout << "How many numbers would you like to enter?" << endl;
    if (cin >> numOfNumbers)
    {
        unsigned int i;

        /* input numbers from console */
        cout << endl << "Please enter " << numOfNumbers << " numbers:" << endl;
        for (i = 0; i < numOfNumbers; i++)
        {
            string input;
            cin >> input;
            numberStrings.push_back(input);
        }

        /* print numbers out */
        cout << endl << "You entered:" << endl;
        for (const auto& num: numberStrings)
        {
            cout << num << endl;
        }
    }
    else
    {
        cout << "Number of numbers must be a positive integer less than " << UINT_MAX << endl;
    }

    return 0;
} /* vector magically deallocates when it leaves scope */

Tangentally, and generally-unrelated to the above: "and if you need to store 'anything' as a string," you can always use JSON, YAML, or XML. These formats are portable, self-describing, and readily available.