You can take these quests because you have class that can take them but you can't turn them in because your current class doesn't have high enough lvl.
It happened to me when I was lvling archer for BLM job. Since I had THM lvl 35 at that time, on my low lvl archer I could take any quests my 35lvl THM was able to take, but I obviously couldn't turn them in.
It's good idea to check level when turning in quest, but it would be even better if you couldn't take these quests to begin with if your lvl is too low.