First, determine the specific data you want to extract from The New York Times website. This could be tables, charts, or specific datasets available in articles. Ensure the data is publicly accessible and doesn't require authentication or subscription.

The New York Times provides an API for accessing its data. Visit the [New York Times Developer Network](https://developer.nytimes.com/) and sign up for an API key. Explore the API documentation to understand how to query the data you need.

Using the API documentation, construct the URL for the API request. This URL will include your API key and any specific parameters needed to filter and retrieve the desired data from The New York Times API.

Open Google Sheets and navigate to `Extensions` > `Apps Script`. Write a script using Google Apps Script to fetch data from The New York Times API. Use the `UrlFetchApp.fetch()` method to send the request to the API endpoint you constructed. Here is a basic example:

```javascript
function fetchData() {
const apiKey = 'YOUR_API_KEY';
const url = 'YOUR_API_REQUEST_URL';
const response = UrlFetchApp.fetch(url);
const data = JSON.parse(response.getContentText());
return data;
}
```

Once the data is fetched, it will typically be in JSON format. Parse this JSON data within your script to extract relevant information. Organize the data into a structure that matches how you want it to appear in your Google Sheet.

Use the `SpreadsheetApp` service in Google Apps Script to insert the parsed data into your Google Sheet. You can specify the starting cell and ensure the data is written in a tabular format. Here's an example of how to write data to a sheet:

```javascript
function writeToSheet(data) {
const sheet = SpreadsheetApp.getActiveSpreadsheet().getActiveSheet();
const dataArray = []; // Convert your parsed data into a 2D array
// Populate dataArray with your data
sheet.getRange(1, 1, dataArray.length, dataArray[0].length).setValues(dataArray);
}
```

To keep your data updated, set up a trigger in Google Apps Script. Go to the Triggers section (`Triggers` > `Current project's triggers`) and create a new trigger for the `fetchData` function. You can schedule it to run at regular intervals, such as daily or hourly, based on your needs.

By following these steps, you can successfully pull data from The New York Times into Google Sheets without using any third-party connectors or integrations.