What did Sherlock Holmes do during his three years absence?

Answer 1

This is the account told to Watson by Holmes in 'The Adventure of the Empty House':

". . . I took to my heels, did ten miles over the mountains in the darkness, and a week later I found myself in Florence, with the certainty that no one in the world knew what had become of me.

I had only one confidant - my brother Mycroft. I owe you many apologies, my dear Watson, but it was all-important that it should be thought I was dead, and it is quite certain that you would not have written so convincing an account of my unhappy end had you not yourself thought that it was true. Several times during the last three years I have taken up my pen to write to you, but always I feared lest your affectionate regard for me should tempt you to some indiscretion which would betray my secret. For that reason I turned away from you this evening when you upset my books, for I was in danger at the time, and any show of surprise and emotion upon your part might have drawn attention to my identity and led to the most deplorable and irreparable results. As to Mycroft, I had to confide in him in order to obtain the money which I needed. The course of events in London did not run so well as I had hoped, for the trial of the Moriarty gang left two of its most dangerous members, my own most vindictive enemies, at liberty. I travelled for two years in Tibet, therefore, and amused myself by visiting Lhassa and spending some days with the head Llama. You may have read of the remarkable explorations of a Norwegian named Sigerson, but I am sure that it never occurred to you that you were receiving news of your friend. I then passed through Persia, looked in at Mecca, and paid a short but interesting visit to the Khalifa at Khartoum, the results of which I have communicated to the Foreign Office. Returning to France, I spent some months in a research into the coal-tar derivatives, which I conducted in a laboratory at Montpelier, in the south of France. Having concluded this to my satisfaction, and learning that only one of my enemies was now left in London, I was about to return, when my movements were hastened by the news of this remarkable Park Lane Mystery, which not only appealed to me by its own merits, but which seemed to offer some most peculiar personal opportunities. I came over at once to London, called in my own person at Baker Street, threw Mrs. Hudson into violent hysterics, and found that Mycroft had preserved my rooms and my papers exactly as they had always been. So it was, my dear Watson, that at two o'clock to-day I found myself in my old arm-chair in my own old room, and only wishing that I could have seen my old friend Watson in the other chair which he has so often adorned.

But anyone who knows a bit of history knows that almost everyone of these places would have been off-limits to any European (or any westerner for that matter) even 120 years ago.