Why do we reject turing machines that use space less than the log of the length of the input?
Consider any program in high-level language that has a loop going over all items:

for i from 1 to n
    do something
end for


Implementing this loop takes $O(\log n)$ work space, since the variable $i$ takes $O(\log n)$ bits to store. If you're not allowed to use even that much memory, then you have to be very careful when implementing any non-trivial algorithm, if it is possible at all; for instance, any results will be highly dependent on the exact definition of the model of computation.

Arora and Barak don't want to focus on such issues, so they just assume that you are given at least logarithmic space.  There are enough interesting things to say without having to worry whether some general reduction can be carried out in sublogarithmic space.  (And if you want to study regular languages -- languages that can be recognized with constant space -- you don't need complexity theory.)