It's either your localstorage is full or you are trying to play in incognito/private mode. If you are, please don't. The whole incognito/private reason to exist is to not leave any data on your computer and you don't want to play games in that mode. If you are in a normal session then you have to check your localstorage, each browser have its own limits and a specific folder where it places saves, if you let me know which one you use i'll try to give you some specific help otherwise the suggestion will be a generic free up some localstorage by opening games, export and delete saves.
You can also be more drastic and delete all the localstorage (manually locating the proper folder in you hard drive or using an addon like StoragErazor for Firefox), but be wary that this will lead to the loss of all your savegames so do that only if you don't care about those.