Look at About Opera (from the menu), it will tell you where your Profile folder is. In that folder, you'll see files named Bookmarks, favorites.db, etc. along with various .bak or .dbak and .journal files. You definitely want at least Bookmarks and Bookmarks.journal, but you might as well make copies of everything in your profile just to be safe.
I do see a "Reset synchronization" button in the Sync options dialog, but you have to be logged in to see it so that doesn't help. I wonder whether deleting the files in the Sync Data folder would work?