How long is a sheinat ara'i?

I think that it is less than 30 minutes.

In which Halacha are you referring to? Is it regarding sleeping with T'Filin or sleeping in Sukah or L'Inyin Birchas hatorah?

Here are some sources:

Tefillin
Shuchan Aruch: 44:1 As long as tefillin are on his head or on his arm, one is forbidden to sleep even a momentary nap wearing them. On the other hand, if one put a cloth over the tefillin and he does not have his wife with him, he may sleep a momentary sleep wearing them.

Mishna Berura-4: A momentary sleep: There is no maximum duration for the sleep to be considered as such. There are authorities who say that there is such a maximum duration and it is as long as it takes to walk a hundred cubits, which is approximately a sixty-seventh of an hour. The Gra rules in accordance with this latter opinion in his Beyur.

Sukkah
Shuchan Aruch 639:2 One should eat and drink and sleep in the sukkah on all seven days of Sukkos, both by day and by night. One may not sleep outside the sukkah, even a short nap.

Mishna Berurah -11: Short nap: This is because a short snooze suffices for a person and is like proper sleep. See Sec. 44 , sub-Par. 4 of the Mishnah Berurah, that the period of time required for a nap is the time that it takes to walk a hundred cubits. Sleep for less than that length of time does not even constitute a nap. It may be that the same ruling applies with respect to the requirement of sleeping in the sukkah, that sleep for that length of time is what is considered snoozing. [P.Mg.] If someone fell asleep outside the sukkah through compelling circumstances, he must go to the sukkah as soon as he wakes up. This is because even when rain fell and then ceased one is required to go to the sukkah if he has not yet lain down, as the M.A. writes below in Par. 7.


Birchas Hamazon

Shuchan Aruch 178:7 If a person fell asleep during the meal for a short nap it is not considered as an interruption, and he does not need to make another blessing when he resumes eating.

Mishna Berura-48: For a short nap: Even if one delayed continuing his meal for a length of time of about an hour, it is nevertheless not considered as an interruption of the meal, since he was overcome by sleep. Consequently, he does not need to make the blessing Ha-Motzi again.


Birchas Hatorah

47:11 A settled sleep in the daytime on one’s bed is ruled as an interruption and one must therefore repeat the blessings if he wishes to study Torah subsequently. However, there are authorities who say that it is not considered as an interruption and the custom in fact follows this opinion.

Mishna Berurah- 25: And the custom in fact follows this opinion: The author of the L. Ch. writes, “It appears to me that if one says the blessings in such a case he will receive blessing and it was in fact the custom of my teacher Mahari to say the blessings in these circumstances.” The Pr.Ch. and the E.R., who bases his conclusion on many Rishonim and Acharonim, also agree with this ruling. The Gra likewise writes this and it is what the Ch.A. notes as the halachic ruling. [Even though it is implied by the P.Mg. that one may rely on the custom described in the Shulchan Aruch not to say a blessing in these circumstances, it at any rate appears obvious that if one relies on all the Poskim we have mentioned and says the blessings he will not lose by it.]



According to the Gra that the Mishna Brurah quoted, "a nap" (shinas ari), is the time it takes to walk 100 amos, which is less than a minute, but I could not find the Gra inside. From what I understand from these sources, this short period of sleep only applies to Tefillin and Sukkah.

what about the need to wash one's hands? Let's say one slept sheinat ara'i, how long does he need to sleep in order to require that his hands be washed?