One can certainly make the basic definitions, and the real issue is to show that the definition "works" using any etale map from a scheme. More precisely, the real work is to show that a weaker definition actually gives a good notion: rather than assume representability of the diagonal, it suffices that $R := U \times_S times_X U$ is a scheme for some scheme $U$ equipped with an etale representable map $U \rightarrow X$. Or put in other terms, we have to show that if $U$ is any scheme and $R \subset U \times_S times U$ is an 'etale subsheaf which is an etale equivalence relation then the quotient sheaf $U/R$ for the big etale site actually has diagonal (over $S$) representable in schemes. Indeed, sometimes we want to construct an algebraic space as simply a $U/R$, so we don't want to have to check "by hand" the representability of the diagonal each time.
That being done, then the question is: which objects give rise to a theory with nice theorems? For example, can we always define an associated topological space whose generic points and so forth give good notions of connectedness, open behavior with respect to fppf maps, etc.? (The definition of "point" needs to be modified from what Knutson does, though equivalent to his definition in the q-s case.) The truth is that once the theory is shown to "make sense" without q-s, it turns out that to prove interesting results one has to assume something extra, the simplest version being the following weaker version of q-s: $X$ is "Zariski-locally q-s" in the sense that $X$ is covered by "open subspaces" which are themselves q-s. This is satisfied for all schemes, which is nice. (There are other variants as well.)
In the stacks project of deJong, as well as the appendix to my paper with Lieblich and Olsson on Nagata compactification for algebraic spaces, some of the weird surprises coming out of the general case (allowing objects not Zariski-locally q-s) are presented. (In that appendix we also explain why the weaker definition given above actually implies the stronger definition as in Chris' answer. This was surely known to certain experts for a long time.)