Magento – How to Override Core Checkout Cart Controller

You need to add config.xml:

<?xml version="1.0"?>
<config>
<modules>
    <SiDev_ModalContacts>
        <version>0.1.0</version>
    </SiDev_ModalContacts>
</modules>

<frontend>
    <routers>
        <modalcontacts>
            <use>standard</use>
            <args>
                <module>SiDev_ModalContacts</module>
                <frontName>modalcontacts</frontName>
            </args>
        </modalcontacts>
    </routers>
</frontend>
 <global>
    <rewrite>        
        <sidev_modalcontacts_contacts_indexcontroller>
            <from><![CDATA[#^/contacts/index/#]]></from> <!-- Mage_Contacts_IndexController  -->
            <to>/modalcontacts/contacts_index/</to> <!-- SiDev_ModalContacts_Contacts_IndexController  -->
        </sidev_modalcontacts_contacts_indexcontroller>
    </rewrite>
    </global>
</config> 

Your controller code as below:

app\code\local\SiDev\ModalContacts\controllers\Contacts\IndexController.php

<?php
require_once "Mage/Contacts/controllers/IndexController.php";  
class SiDev_ModalContacts_Contacts_IndexController extends Mage_Contacts_IndexController{
 //your code goes here
}

First of all...I advice against rewriting controllers.
Actually I advice against rewriting anything if you can do it an other way.
Second: The class Magento\Checkout\Controller\Cart is an abstract class. It is never instantiated so you cannot rewrite it.
I assume you want to replace the behavior of the execute method from one of the classes that extend Magento\Checkout\Controller\Cart.
If so, you can do it with an around plugin for your execute method in the original controller.
Let's take for example Magento\Checkout\Controller\Cart\Add controller action

For this you will need to add this to the di.xml of your module:

 <type name="Magento\Checkout\Controller\Cart\Add">
    <plugin name="buy-x-get-y" type="StwBuyXGetY\BuyXGetY\Plugin\Controller\Checkout\Cart\Add" sortOrder="1" />
</type>

Then create the file StwBuyXGetY/BuyXGetY/Plugin/Controller/Checkout/Cart/Add.php with this content:

namespace StwBuyXGetY\BuyXGetY\Plugin\Controller\Checkout\Cart;
class Add
{
    public function aroundExecute(\Magento\Checkout\Controller\Cart\Add $add, \callable $proceed)
    {
        //your code goes here
        //if you don't want the original method to be called ignore what'e below.
        $returnValue = $proceed();
        return $returnValue;
    }
}

I haven't tested the code, so look out for typos.