This is likely due to a failed auto-update.
There are steps to take:
Backup all site files and db first using FTP and host tools.
1. Perform a Manual Update - this requires FTP or other host provided access to the files. Remove the two folders wp-admin and wp-includes - do not touch or delete wp-content folder at this point. Download a fresh copy of WordPress. Extract the files and upload only these two folders. Do not access site until all steps are taken.
2. Rename the plugins folder /wp-content/plugins/ to /wp-content/old-plugins. This deactivates any plugins that may be causing an issue.
3. Rename all other theme folders found in /wp-content/themes/ to old-theme_folder_name. Upload a current version of the default theme (twentytwelve - from the current WP download you made in step 1.)
4. Rename the .htaccess file found in the root folder of your install to old-htaccess.
5. Attempt to login - you will now have a default install up and running. Once logged in run the site to see that it runs - the theme and plugin folders can be renamed back and then activated to test. Prior to doing so verify that each is compatible with WP3.5.1.