I'm posting this in case anyone else runs into the same problem.
I just installed a fresh copy of WordPress on my server and immediately received:
PHP Fatal error: Call to undefined function get_option() in /var/www/wp-admin/admin.php on line 32
...when trying to access the admin side. I re-downloaded and re-installed at least 5 times and verified the files and functions were there, but it still gave me that error.
After hunting for hours and not finding any solutions, it turns out the problem was that I had
apc.include_once_override set to 1 in my server's configuration. After setting it to 0, everything worked wonderfully again.
I hope this saves someone some time.