The Gemara relates that such a Succah i.e. a 7x7 tefachim Succah can still hold ראשו ורובו ושלחנו - his head, the majority of his body and his table - Refer also to the Mishnah Berurah 634:1.
Such dimensions did still allow for sleeping. Bear in mind, as Rashi asserts (also mentioned in Sh'ar HaTzion 634:1), "דרך סעודתן בהסיבה" they would eat in those days leaning, so they were anyway half lying down to a certain extent.
ראשו ורובו - דרך סעודתן בהסיבה היתה על מטות ומסובין על צד שמאלם ואין אוכלין זקופין ויושבין כמונו לפיכך הוצרך לומר ראשו ורובו לאפוקי קטנה שראשו נכנס בה ושלחנו ולא רובו או ראשו ורובו ולא שלחנו
His head and the majority of his body - the way of eating their meal was by leaning on beds (chaise-longues?) and leaning on their left side, and they wouldn't eat upright, and they would sit like this. Therefore it is necessary to say "his head and the majority of his body" to exclude where a little part of his head went into it and his table but not the majority of his body, or his head, the majority of his body but not his table.
So whilst possibly uncomfortable, one is still able to sleep in such a Succah - Refer to the diagram printed at the bottom of the page in the Peirush Chai here