I think I finally fixed it.
This stupid “OMG Nepomuk is not running!” error has been dogging me every time I launch KMail, ever since Fedora upgraded KDE. I followed all the directions on fixing Akonadi, and nothing worked. Finally, it turned out that there was a config file telling it to load the old-style “redland” database — which doesn’t exist in current versions of Nepomuk — instead of the new “virtuoso” database.
The file was in
~.kde/share/config/nepomukserverrc and the item in question is “Used Soprano Backend.” I changed it from “redland” to “virtuosobackend” as described here and now it actually starts Nepomuk, and KMail doesn’t complain when I start it!
Someone at KDE decided to massively overcomplicate things!