613 commandments and 613 limbs of the body

The following answer is by Prof. Shmuel Kottek:
Today, gid is a tendon , a sinew. However the term is used in ancient Hebrew for all kinds of vessels in the body, nerves, blood-vessels and more. None of them relates to number 365. The Mishnah tries to solve the problem of 248 bones, which is not exact either. Hippocrates had no exact number either.  In fact, there is no answer to your question (so far). 

Note By Rabbi Dr Mordechai Halperin:
Nevertheless, we can understand the enumeration made by the sages if we keep in mind the Halachic perpose of counting the bones.
See:
1. Tractate of Ohalot, a scientific edition, with a commentary and introduction by Abraham Goldberg, Magnes Press, vol. 1, Mishnayot 9-10, and also in commentary on chapter 2, Mishna 1, "And how much… – one hundred and twenty-five" [Note that there is a typographical error there, Bekhorot 45a should be instead of Succah 45a]
2. YL Katzenelson, The Talmud and the Wisdom of Medicine, published by Haim Publications, Berlin 1928 (Photo Edition, Twersky, Tel Aviv, 1982, pp. 254-303)
3. In addition, see  'bones and ribs' in Encyclopedia of Jewish Medical Ethics, vol. I p. 119-120  
4. See Julius Preuss, Biblical & Talmudic Medicine, (trns. ed. Dr. Fred Rosner, Pub. Jason Aronson inc., Northvale-New Jersey-London, 1994), p. 60. Pruess brings the Aggadic lore as well as the Halachic reasoning and the variations in numbering the limb numbers.