The Gemara in Rosh ha'Shanah (21a) cites Rava (in whose time the Sheluchim still used to arrive from Eretz Yisrae), who used to fast on the tenth and eleventh of Tishri, due to S'feika de'Yoma. Then it cites Rav Nachman, who used to fast only on the tenth, and in the course of an episode that occurred there he claimed that fasting two days was life-threatening.

Rabeinu Chananel explains that what Rava did (according to him it was Rabah) was Midas Chasidus and not obligatory.

No doubt the reason that Rav Nachman fasted specifically on the tenth was based on the fact that Elul was hardly ever Me'ubar (though the Gemara does not rely on that with regard to all the other Yomim Tovim).

Consequently, the Chasam Sofer (and other Poskim) rule that nowadays, it is futile to fast two days (in keeping with the Minhag on all other Yomim Tovim). The reason for that is because we only observe two days Yom Tov since that is what our ancestors used to do. However with regard to Yom Kipur, never, at any time, did they fast two days.

Eliezer Chrysler,

Kollel Iyun Hadaf