The Gemara in Masechet Shabbat (69b) addresses the situation of a person who travels through an uninhabited area and loses track of the days of the week. When, and how, does a person observe Shabbat in such a situation?
Halacha requires an individual in this case to count six days starting from the day he lost track of the days, and then observe Shabbat on the seventh day. Even though we have no reason to assume that his seventh day is Shabbat, he should nevertheless observe that day as Shabbat. He should recite Kiddush and Habdala, and refrain from activity that is forbidden on Shabbat.
However, as the Shulhan Aruch rules, this person must still take into account the possibility that any given day is actually Shabbat. Meaning, even during the six days when Halacha allows him to perform Melacha (activity forbidden on Shabbat), he may not do more than is needed for securing his basic survival. The principle of "Piku’ah Nefesh" allows (and even requires) violating Torah law for the purpose of sustaining human life, and therefore a person in the situation described is allowed to work during the six days despite the possibility that any given day is Shabbat. However, he may not work beyond that which is necessary for his basic necessities.
By the same token, on the seventh day, which he observes as Shabbat, he is allowed and required to perform Melacha if this is necessary for ensuring his survival. Preferably, he should make adequate preparations on the sixth day so that he would not have to work on the seventh. But if circumstances necessitate working on the seventh day to ensure his survival, then he should certainly do whatever is needed to sustain himself.
What prayers does he recite on the seventh day, or, for that matter, during the six days?
The Hid"a (Rav Haim Yosef David Azulai, 1724-1806) ruled that he should recite the weekday prayer service every day, including the seventh day. A person who recites the weekday prayer service on Shabbat fulfills his obligation of prayer, despite the fact that he did not recite the special Shabbat prayer text. Therefore, in a case where a person does not know whether that day is a weekday or Shabbat, it is preferable to recite the weekday prayer service, as he thereby ensures to fulfill his obligation. Moreover, since he is required to recite Kiddush, he will be reciting a special text for Shabbat despite his reciting the weekday prayer. The Hid"a also writes that this individual should not recite Musaf, even on the seventh day, since he is uncertain whether this day is Shabbat.
Does this individual wear Tefillin on the seventh day?
This issue is subject to debate among the Halachic authorities. The Kaf Ha’haim (Rav Yaakov Haim Sofer, Baghdad-Israel, 1870-1939) cites from the work "Shabbat Shel Mi" that a person in this situation wears Tefillin every day, including the seventh day, which he observes as Shabbat. Others, however, maintain that he wears Tefillin only during the six days, and even on those days he does not recite the Beracha over the Tefillin.
Tosafot (Talmud commentaries by Medieval French and German scholars) comment that it is permissible for this individual to travel, even on the seventh day, which he observes as Shabbat. Since we want him to leave these circumstances as soon as possible, rather than remain in a position where he does not know the day of the week, he may travel every day so he could quickly make his way back home.
The Radbaz (Rabbi David Ben Zimra, Egypt, 1480-1574), in one of his responsa, discusses the case of an individual who had lost track of the days while traveling, and then, upon returning home, discovers that he had observed the wrong day as Shabbat. According to the Radbaz, if this happened in the times of the Bet Hamikdash, the individual would not be required to bring a Korban Hatat (sin offering) for his inadvertent violations of Shabbat. Nevertheless, he should perform Teshuba (repentance).
If a person happens to be in the North Pole or South Pole, where it is daytime or nighttime consecutively for six months, how does he determine when to observe Shabbat? Normally, we count six days based upon the rising and setting of the sun. But how does one know when to observe Shabbat in a place where the sun does not rise, or does not set, for months at a time?
The Hid"a rules that in such a case, a person simply counts six periods of twenty-four hours, and then observes Shabbat during the seventh twenty-four-hour period. He recites Kiddush at the beginning of this period, and recites Habdala at its conclusion.
Summary: If a person is in an uninhabited region and loses track of the days, he should count six days from the day he loses track and observe the seventh day as Shabbat. However, he recites the weekday prayer on the seventh day, and does not recite Musaf. According to some views, he wears Tefillin on the seventh day. On the other six days, he should perform only the work he needs to sustain himself, given the possibility that one of those days is Shabbat. In places where the sun does not rise, or does not set, for months at a time, one should count six periods of twenty-four hours and then observe Shabbat during the seventh twenty-four-hour period.
