I have editors uploading images the size of 3.8MB. I don't want to lower the maximum size for image/file upload (it's set to be 5MB), that's because I use core Image Styles (admin/config/media/image-styles), but for a server's sake, is it possible to make them upload that enormous file but just store a resized, smaller version of it? let's say, store a reduced 800x800px version of it?
Yes, that functionality is in Drupal core as default.
In Image field settings (your Content type -> Manage fields) set for Maximum image resolution to 800x800px. Drupal will resize image size and save only resized version of it.