In Archwiki has a lot of useful information. Dropbox: Known Issues
Your case is similar to this problem:
Dropbox does not start — "This is usually because of a permission error"
Errors caused by running out of space
A common error that might happen is that there is no more available space on your /tmp and /var partitions. If this happens, Dropbox will crash on startup with the following error in its log:
Exception: Not a valid FileCache file
A detailed story of such an occurrence can be found in the forums
. Make sure there is enough space available before launching Dropbox.