This does seem like a corrupt WP core with missing files. You can attempt to repair it by uploading a fresh copy of core in place. The procedure would be along the lines of Manual Update documented in Codex.
However I want to note that pieces of WordPress core don’t just disappear for no reason. You might have been hacking, affected by server issues, or something else entirely. The request to hosting support to look into situation would definitely be wise.
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