To store objects in HTML5 localStorage or sessionStorage, you need to first convert the object to a string using JSON.stringify() and then store it as a string. When you want to retrieve the object, you can use JSON.parse() to convert the string back to an object.

Here's an example:
Storing an object in localStorage

const myObject = {name: "John", age: 30};
localStorage.setItem("myObject", JSON.stringify(myObject));

 Retrieving the object from localStorage
const storedObject = JSON.parse(localStorage.getItem("myObject"));
console.log(storedObject); // {name: "John", age: 30}

Similarly, you can store objects in sessionStorage using the same approach:

Storing an object in sessionStorage
const myObject = {name: "John", age: 30};
sessionStorage.setItem("myObject", JSON.stringify(myObject));

Retrieving the object from sessionStorage
const storedObject = JSON.parse(sessionStorage.getItem("myObject"));
console.log(storedObject); // {name: "John", age: 30}

Keep in mind that localStorage and sessionStorage have a size limit, which varies by browser. Also, it's important to note that objects stored in localStorage or sessionStorage are not secure and can be accessed by anyone who has access to the browser.