What is a space constructible function?
The space used by a Turing machine is the total number of cells that the head (or heads, if a multitape machine) of the Turing machine has ever visited. A function $f(n)$ is space-constructible if there is a Turing machine that on input $1^n$ outputs the binary representation of $f(n)$, and while doing so only visits $O(f(n))$ cells. Note that $O(f(n))$ is an upper bound on the visited cells, not on the length of the representation.

Whether constant functions are space-constructible or not according to this definition depends on the Turing machine model. If the input and output are on the same tape, then a Turing machine space-constructing $c$ should first erase the input $1^n$ and then write the binary representation of $c$. Unfortunately, this takes $n$ space rather than $O(c) = O(1)$ space. If, in contrast, the input is on a special read-only tape and the output is on a special write-only tape (as is the case for log-space machines), then there is absolutely no problem to (fully) space-construct $c$.