Firebase Storage have built-in error handler. See example code below:

import { getStorage, ref, getDownloadURL } from "firebase/storage";

// Create a reference to the file we want to download
const storage = getStorage();
const storageRef = ref(storage, `profiles/${_authContext.currentUser.uid}/profile-image`);

// Get the download URL
getDownloadURL(storageRef)
  .then((url) => {
    if (url) {
      setLoaded(url)
    }
  })
  .catch((error) => {
    // A full list of error codes is available at
    // https://firebase.google.com/docs/storage/web/handle-errors
    switch (error.code) {
      case 'storage/object-not-found':
        // File doesn't exist
        break;
      case 'storage/unauthorized':
        // User doesn't have permission to access the object
        break;
      case 'storage/canceled':
        // User canceled the upload
        break;

      // ...

      case 'storage/unknown':
        // Unknown error occurred, inspect the server response
        break;
    }
  });

The code above should handle the error from Firebase. If the file doesn’t exist, there’ll be a GET 404 (Not Found) error in your console / network tab. There's a workaround for this scenario, you can run a list() or listAll() command to ensure if the file exist in the directory before running the getDownloadURL(). See sample code below for reference:

useEffect(() => {
         const getImg = async () => {
           const storage = getStorage();
           const storageFolderRef = ref(storage, `test/`);
           const imageRef = ref(storage, `test/profile-image.png`);
          // Find all the prefixes and items.
          listAll(storageFolderRef)
            .then((res) => {
              if (res.items.length > 0) {
                getDownloadURL(imageRef)
                .then((url) => {
                  if (url) {
                   setLoaded(url)
                  }
                })
                .catch((error) => {
                 // A full list of error codes is available at
                 // https://firebase.google.com/docs/storage/web/handle-errors
                 switch (error.code) {
                   case 'storage/object-not-found':
                     // File doesn't exist
                     break;
                   case 'storage/unauthorized':
                     // User doesn't have permission to access the object
                     break;
                   case 'storage/canceled':
                     // User canceled the upload
                     break;
             
                   // ...
             
                   case 'storage/unknown':
                     // Unknown error occurred, inspect the server response
                     break;
                 }
                });
              }
            }).catch((error) => {
              console.log(error);
            });
         };
         getImg();
     }, []);