Robin Hood is, among other things, the titular character in "Robot of Sherwood." Is he a robot? Um...maybe? Probably not. But what he is is absolutely everything you expect from Robin Hood - green-clad, dashing, laughing in the face of danger, unrivaled in archery, noble thief, the lot. Which, honestly, is totally unexpected. He is Robin Hood the Legend, by contrast to, say, Hal the Archer in "the Time Warrior," who has much more of the feel of a historical Robing Hood analogue (although Hal, of course, is not an outlaw). The whole feel of the episode is very much
Ye Goode Olde Days, and not at all
the Dung Ages. But it's not just the perfect hair and teeth and cleanness - Robin is simply too Robin-Hood-y to be real.
The tension between fantasy and reality and the role of stories in mediating between the two has been the major theme of the most recent era of Doctor Who. The Doctor takes the existence of Robin Hood as a personal offense. He has seen first hand the dangers of becoming a story - being seen as a Hero nearly destroyed him and the people he cared about, and this mythification is one of the past mistakes he is trying to correct. His arc this season is one of casting off the legends that have overtaken him and trying to figure out what, if anything, he is underneath - culminating in the epic "I'm an idiot!" reveal. But Robin is doing precisely the opposite - he has embraced his legend, actively seeking to be the Impossible Hero that the stories make him.
What is important to me about Roin Hood is that he is very much playing a role, but it is the role he has chosen for himself. He seems not quite real, not because he is a killer robot (probably) but because he is actively trying to be larger than life. He laughs and banters and makes horible puns and walks into the Obvious Trap - but he laughs too much, and he watches and learns - applying, for instance, the Doctor's winning spoon-fighting move. He's much much cleverer than he presents himself (which occasionally adds to the overall impression that he's actually a killer robot after all). His cheery affect comes not from the fact that his life is fairy-tale perfect but rather from the fact that it isn't. And so "Robin Hood" is something that he does, because not only is the fantasy better than reality, it makes reality better. His attempt to steal the TARDIS is ridiculous, but it very skillfully makes a specific impression - it's what Robin Hood (TM) would do. In the words of Zaphod Beeblebrox: "So, ten out of ten for style, minus several million for good thinking, hey?" But Robin understands that sometimes the style really is more important than the good thinking, provided you can stick the landing. And of course the good thinking is still happening, just very quickly and behind the mask, where it's all the more unexpected.
The one time he drops his mask is when he questions Clara about Robin Hood, and about the Doctor. We don't know what exactly she tells him, but surprisingly it seems to be the truth, and perhaps even more surprisingly he believes every word of it. And tries to help. That actually says an awful lot - when confronted with the reality of time travel and aliens and distant worlds and killer robots, what Robin chooses to do is validate the Doctor. In the end he shows him what he wants to see - that it is an act - but he also defends the idea of storyhood. Robin tells the Doctor that it is okay to be a Legend and a Hero, something the Doctor is currently running from just as fast as he can. But Robin Hood is the mythification success story and poster child, and is very happy to know it. "History is a burden - stories can make us fly." Storyhood to Robin means the ability to be all things to all people - to be what is needed in every case, without fear of being remembered for his failures and foibles. It doesn't matter whether he's a man or a robot - history has forgotten both of them. His parting line is "I'm just as real as you are" with everything that means one way and the other. Both of them exist, both are ridiculous and impossible, and ultimately the question of reality or not doesn't matter. What matters is what they mean to other people.