Here is the issue from my debugging, for future reference. The server I am working on apparently has a package called eAccelerator for PHP. Laravel appears to be incompatible with this PHP accelerator, and it breaks it in unpredictable ways.
Add to the very top of public/index.php:
ini_set('eaccelerator.enable', 0);
For what its worth, I then got a blank screen, which was solved by changing the permissions of the storage folder: http://stackoverflow.com/questions/20678360/laravel-blank-white-screen
The command in 5.0:
chmod -R guo+w storage
I really wish it were possible to give an error like "The permissions don't allow Laravel to write anything to the storage folder, which it needs to do to load your application. Please set the permissions to..."
I have had blank screens with no feedback on almost every one of my new laravel projects on different servers.
Listen, I think Laravel is amazing. But nothing makes Taylor's peppy, cheery comments like "It feels nice to relax" more annoying than a blank screen with no output or error log on a fresh install.