Looking for an example of proving space upper bounds for computing functions on a DTM
Perhaps the simplest way to show that a certain task can be done with small space is writing C-like pseudocode for it. If you are only using a constant number of variables, and each variable is exponentially bounded in terms of the input length (i.e. $|x| = O(k^n)$ for fixed $k$), then the resulting computation uses at most linear space. You can use the fact that addition, multiplication and division can be done in linear space ($O(n)$), as well as operations like determining the input length.

As an example, here is pseudocode for your algorithm:

f(x) {
  l = |x|;
  y = l * l;
  for i from 1 to y do:
    output 0;
}


The variables here are $l,y,i$ ($x$ is on the input tape so doesn't count). All operations are in linear space (as long as the inputs are small enough), $l \leq n$ and $y,i \leq n^2$, so the algorithm uses linear space.

If the bounds on the variables are polynomial ($O(n^k)$) rather than exponential ($O(k^n)$), then the resulting program is logspace (uses $O(\log n)$ space); all operations mentioned can be implemented in logspace. Indeed, since the bounds in your case are polynomial, your algorithm is logspace; the fact that the output is larger doesn't matter, since we only count space on the work tape.