Just as I am surprised that a couple of episodes that I really enjoyed are rated so low, I am surprised that "Elaan of Troyius" isn't generally held in lower regard. In my opinion, it is one of the weakest and dullest episodes of the entire Original Series.
The plot reminds me of the type that we would often see from 'The Next Generation' onwards - entirely ship-bound, and not all that much of interest happens. The fact that it comes from the Original Series, which was usually more adventurous with its settings, makes it seem even weaker.
I couldn't care less about spoiled princess Elaan, and found that the episode really dragged. And ... would you believe it ... she falls for Captain Kirk. Well there's a plot twist that we've never seen before(!)
The Klingon b-plot does nothing to even things up, and I found the solution, involving Elaan's necklace of all things, to be over-convenient and unconvincing.
Not much more to say on this one. It defiantly ranks in my Top (or should that be Bottom?) Five least favourite episodes of the series. This is another one that I'd say, for the completists, sit through it once to say you've seen it, and then move on.