In the new Doctor Who episode, the Doctor meets Robin Hood, and it's very clearly Robin Hood as he is known in modern media, specifically an Errol Flynn/Douglas Fairbanks-style Robin Hood. (Errol Flynn even gets name-checked, and there's a fun part where the Doctor is reviewing stories about Robin Hood and we see the actor who played the second Doctor, Patrick Traughton, from the 1950's TV show "The Adventures of Robin Hood".) Which is exactly why the Doctor doesn't believe he's really Robin Hood for most of the episode and thinks he must be an impostor.
When it turns out he IS Robin Hood, it's not actually the show saying that this is who the historical Robin Hood was, because that's clearly malarky. The Robin Hood of the original ballads was not this person, and even those were from long after Robin was supposed to have lived and grew out of a very particular folk hero tradition. No, what Doctor Who is saying is that the television show Doctor Who doesn't actually travel to the past at all, but travels to popular impressions of the past, the past as it is imagined. The past as a genre, as a narrative trope that Doctor Who can then crash into, interrogate and subvert. Which is why Robin Hood tells the Doctor "Remember, I'm just as real as you are."
Which is essentially a call-back to a second Doctor adventure called the "The Mind Robber" (1968), where the Doctor and his companions get lost in the Land of Fiction, a world populated by fictional characters. And one of the storybook characters tells the Doctor that he's a "traitor" to the Land of Fiction. As if he was a part of it that rebelled. (See also Philip Sandifer's brilliant analysis of "The Mind Robber".) At the end, it's not even clear that they ever really escaped the Land of Fiction.
Thus Doctor Who becomes a show aware of it's status as a work of fiction. Doctor Who always gave us the Doctor crashing into one or more other genres, but "The Mind Robber" gave that fact context. And so, much as Tarantino makes a WWII movie that's less about WWII and more about how WWII has been portrayed in popular culture, Doctor Who makes an episode about Robin Hood that doesn't pretend to be about the historical Robin Hood but instead is about interrogating what Robin Hood means to us and about rescuing us from our cynicism towards heroes. Because for someone from the Land of Fiction, believing in Robin Hood has nothing to do with history and everything to do with what Robin Hood means to us.
This isn't to say this is the greatest episode of Doctor Who, but it's one I enjoyed a lot. And anyone complaining that this isn't historically accurate doesn't understand this show.
PS. Yes, the shooting the golden arrow into the literal bulls-eye on the ship is silly. It's a silly show. The Doctor fights Robin Hood with a spoon. Get over it.