You can export images, map tiles, tables and video from Earth Engine. The exports can be sent to your Google Drive account, to Google Cloud Storage or to a new Earth Engine asset.
To use Google Cloud Storage (a fee-based service), you'll need to set up a project, enable billing for the project, and create a storage bucket. See the Cloud Storage Quickstart page for instructions. See this guide for information on storage bucket naming. Data exported to a Cloud Storage bucket will have the bucket's default object Access Control List (ACL). You must have write permission for the specified bucket.