localStorage.setItem() and localStorage.getItem() can be used to Get and Set localStorage variable in Salesforce Lightning Web Component.

Sample Lightning Web Component:

HTML:

<template>
    <lightning-card>
        <div class="slds-var-p-around_medium">
        {strValue}
        </div>
    </lightning-card>
</template>

JavaScript:

import { LightningElement } from 'lwc';

export default class sampleLightningWebComponent extends LightningElement {

    strValue;

    constructor() {

        super();
        localStorage.setItem( 
            'exampleItem', 
            'Testing' 
        );
        console.log( 
            'exampleItem is', 
            localStorage.getItem( 
                'exampleItem' 
            ) 
        );
        this.strValue = localStorage.getItem( 
            'exampleItem' 
        );

    }

}

js-meta.xml:

<?xml version="1.0" encoding="UTF-8"?>
<LightningComponentBundle xmlns="http://soap.sforce.com/2006/04/metadata">
    <apiVersion>58.0</apiVersion>
    <isExposed>true</isExposed>
    <targets>
        <target>lightning__Tab</target>
    </targets>
</LightningComponentBundle>

Output: