The Mishna seemingly assumes that all sleeping must be done in the sukka; the distinction between qeva and arai that exists concerning eating in the sukka is not mentioned, although it is mentioned in the baraithoth, the outtakes of the Mishna, and the conclusion of the Gemara is that indeed, although we can make such a syntactical distinction between qeva and arai, one that would matter with regard to certain laws other activities, including sleeping while wearing t’fillin, in practice and with regard to sleeping in the sukka, there is no such distinction (BT Sukka 26-27). It is no surprise therefore that the codes say that sleeping in the sukka is an obligation during the festival, even if one were to want to sleep for a short period of time outside of the sukka.
(None of this touches on the various dispensations that exist for those who for whatever reason are exempt from sleeping in the sukka due to discomfort, etc.)
Why would sleeping in the sukka be held to a higher standard than eating? Why did the sages assume such importance specifically with regard to sleeping in the sukka?
The answer, I believe, is based on ideas expounded upon earlier in this space. Being that s’chach represents the Clouds of Glory, we can then ask where else or what else served as a base for the Clouds of Glory? In the classic literature, the Tabernacles and Temples and the tent of the matriarchs were marked by the Clouds of Glory that hovered over them, symbolizing the true divine protection that made physical roofs and actual walls unnecessary. Just like the Jewish home is a sanctuary, on the the most joyous holiday of the year, every Jew is bidden, if he can not ascend on the pilgrimage to the Temple, to go a step beyond his own domestic sanctuary and build a sanctuary with the closest materials to the Clouds of Glory, and dwell there for the duration of the holiday.
As we saw earlier, “the Guardian of Israel neither sleeps nor slumbers,” and sleep is one inappropriate anthropomorphism that is never used to describe God, but we do find many verses that refer to God resting, as in “He rested on the seventh day,” “where is the place of My rest?” and “arise O Lord to your Your rest, You and Your ark of might,” and “you have yet to come to the resting place and the inheritance,” the latter three being references to the Temple. Similarly, we find the root shin-kaf-nun, to dwell, in words like shechina, the Divine Presence, and many others referring to the Temple.
Further, there are many references to “God’s food.” The sacrifices are called His “bread,” and the priests eat off of the divine table. Yet, although we find that God has only one true resting place where “He causes his presence to reside,” and at any given point in History, God only had only one sanctuary, whether it was Shiloh, or Jerusalem, or elsewhere. However, when the sanctuary was set up in Nov and Gibeon, for example, God did allow some of his sacrifices to be offered outside of the sanctuary on other altars. Indeed, in our sages’s terminology, the sacrifices that had to be offered within those sanctuaries were the ones that had qeva, implying that the ones allowed at the private high places were arai. We therefore have found a precedent showing that even God Himself was more stringent with His resting within His Sanctuary than He was with “eating” within His sanctuary, and the sages may have commuted a similar standard to our performance of the commandment to dwell under the Clouds of Glory.