Step 1: Determine which folder in /var is full (Clean /var)
$> sudo du -csh /var/*
Now you need to dive deeper to figure out the culprits.
Typically the /var/log may be full or contains a lot of log messages. Get rid of the large log files.
Step 2:If your /var/cache/yum is large/full and needs cleanup (Clean /var/cache/yum).
$> sudo yum clean packages
If this does not bring the size of the folder down.
$> sudo yum clean all
Step 3: If you have upgraded Fedora via preupgrade, then your /var/cache/yum/preupgrade folder is large. (Clean /var/cache/yum/preupgrade)
$>sudo rm -fr /var/cache/yum/preupgrade*