The Ubuntu One client on the Mac is a little dodgy in my experience. It sometimes takes several quits and launches for any items selected to sync to the Ubuntu One cloud storage. Do check to make sure that what you have selected to sync through Ubuntu One is within the capacity limit of your Ubuntu One account.
As for why you are not able to upload via a Web browser, I do not know. I have had no issues with it using Safari or Firefox.
Jun 14 16:51:30 Mac-Pro com.apple.launchd (com.ubuntu.one.fsevents): Job appears to have crashed: Segmentation fault: 11
Jun 14 16:51:30 Mac-Pro com.apple.launchd (com.ubuntu.one.fsevents): Throttling respawn: Will start in 8 seconds
Jun 14 16:51:30 Mac-Pro ReportCrash: Saved crash report for com.ubuntu.one.fsevents version ??? (1.3) to /Library/Logs/DiagnosticReports/com.ubuntu.one.fsevents_2013-06-14-165130_localhost.crash