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

Sample Code:

Component:

<aura:component implements="force:appHostable" >
    <aura:handler name="init" value="{!this}" action="{!c.init}"/>
</aura:component>

Component JavaScript Controller:

({
    init : function(component, event, helper) {
        
        localStorage.setItem( 'exampleItem', 'Testing' );
        console.log( 'exampleItem is', localStorage.getItem( 'exampleItem' ) );
        
    }
}) 

Output: