I found it named slightly differently (Win7):
HKEY_LOCAL_MACHINE\SOFTWARE\Policies\Google\Chrome\DeveloperToolsDisabled
I set it to zero and restarted Chrome --> all is well. No idea how it got set.
Update 5/30/2014: This value was set again, there must be some security policy against this flag where I work. I deleted the key completely and will report back if it appears again.
Update 6/13/2014: Definitely we've got a security policy that is setting this key daily. As a workaround I've created a file, "fix_developer_options.reg" with the following contents and put it on my Desktop:
Windows Registry Editor Version 5.00
[HKEY_LOCAL_MACHINE\SOFTWARE\Policies\Google\Chrome]
"DeveloperToolsDisabled"=dword:00000000
This file, when double-clicked, will automatically make this setting in the registry. After that, you need to restart Chrome to get the change. (One step better would be to have this executed when you log in.)
Update 3/4/2018:
In Windows 10 this setting has moved under HKEY_CURRENT_USER as @DGinzberg mentions below in the comments. I found it here on a new computer:
[HKEY_CURRENT_USER\SOFTWARE\Policies\Google\Chrome]