I only accepted the notion of a non-Hausdorff space after I accepted the more general notion of a closure system and closure operators. I think it would make sense for topology textbooks to first define the notion of a closure system or a closure operator, then define the notion of a topological space. After all, if we are not considering any separation axioms, it won't do any harm to generalize the notion of a topological space itself, but instead it will build intuition for an axiomization that seems strange to everyone besides professional mathematicians who tolerate this idea only because they are used to it. Strangely, I have not seen the notion of a closure system discussed in topology textbooks besides the book Beyond Topology.

If $X$ is a set, then a closure operator on $X$ is a function $C:P(X)\rightarrow P(X)$ such that $R\subseteq C(R)=C(C(R))$ and if $R\subseteq S$, then $C(R)\subseteq C(S)$. Intuitively, a closure operator gives one the notion of a closure of a set.

The notion of a closure system is closely related to the notion of a closure operator.
Let $X$ be a set. Then a closure system over $X$ is a subset $C\subseteq P(X)$ such that for each $R\subseteq X$ there is a smallest $S\in C$ with $R\subseteq S$. It is easy to see that a subset $C\subseteq P(X)$ is a closure system if and only if $C$ is closed under arbitrary intersection including the empty intersection. Therefore the notion of a closure system is the notion of the closed sets in a topological space except that the requirement that the union of finitely many closed sets is closed is dropped.

The notions of a closure system and a closure operator are pretty much the same thing.
If $C$ is a closure operator, then define $C^{*}=\{R\subseteq X|C(R)=R\}=\{C(R)|R\subseteq X\}$. Then $C^{*}$ is a closure system. Furthermore, if $C$ is a closure system, then define a mapping $C^{*}:P(X)\rightarrow P(X)$ where if $R\subseteq X$, then $C^{*}(R)$ is the smallest element in $C$ containing the set $R$. Then $C^{*}$ is a closure system. Furthermore, if $C$ is a closure system or a closure operator, then $C=C^{**}$. Therefore the closure systems and the closure operators on a set are in a one-to-one correspondence, so these notions are interchangable. The notion of a closure operator and a closure system can be generalized to a "point-free" setting by considering closure operators on posets (i.e. operators $C:X\rightarrow X$ such that $x\leq C(x)=C(C(X))$ and $x\leq y\Rightarrow C(x)\leq C(y)$).

Closure operators and closure systems abound in all areas of mathematics. For example, the convex sets of a real vector space form a closure system where the closure operator is simply the convex closure that we are all familiar with. The closed sets in a topological space always form a closure system. Also, you get a closure system every time you consider subsets closed under certain operations. The subgroups of a group, the normal subgroups of a group, the subrings of a ring, the ideals in a ring, and the subspaces of a vector space all form closure systems. In fact, the collection of all closure systems on a set $X$ is itself a closure system. From these examples, the intuition behind the notion of a closure operator and a closure system should be clear. Closure operators simply take a set and expand that set until this set cannot be expanded any more and the closed sets are the sets that cannot be expanded further. By the above examples, one can see that this expansion process can be connecting pairs of points by a line segment, adding algebraic combinations of points, adding limits of points, or any combination of these closure systems.

Once one accepts the notion of a closure system, it is easy to accept the notion of a topological space since topological spaces are simply special kinds of closure operators. More specifically, we define a topological closure system to be a closure system $C$ that is closed under finite union (including the empty union). A topological closure operator is a closure operator $C$ such that $C(R\cup S)=C(R)\cup C(S)$ and $C(\emptyset)=\emptyset$. It is easy to show that a closure system $C$ is a topological closure system if and only if $C^{*}$ is a topological closure operator. From this correspondence between topological closure systems and topological closure operators, we immediately obtain Kuratowski's closure operator axioms. Intuitively, the notion behind a topological closure operator or a topological closure system is an expansion process that expands the sets in such a way that the expansion of the union of two sets gives you nothing more than simply the expansion of the two sets individually.

The notion of a continuous function also makes sense in terms of closure systems. If
$(X,C),(Y,D)$ are closure systems, then a function $f:X\rightarrow Y$ is said to be continuous if $f[C^{*}(R)]\subseteq D^{*}(f[R])$ for each $R\subseteq X$.

$\mathbf{Proposition}$
Let $(X,C),(Y,D)$ be topological closure systems, and let $f:X\rightarrow Y$. Then the following are equivalent.

$f$ is continuous.

if $S\in D$, then $f^{-1}[S]\in C$.

if $S\subseteq Y$, then $C^{*}(f^{-1}[S])\subseteq f^{-1}[D^{*}(S)]$.\\

Closure systems also have a nice interpretation in terms of ordered sets and lattices.
If $C$ is a closure system, then define a preordering on $C$ where $x\leq y$ if and only if
$x\in C^{*}(y)$ if and only if $C^{*}(x)\subseteq C^{*}(y)$. A closure system $(X,C)$ is said to be $T_{0}$ if whenever $x,y\in X$, then there is some $R\in C$ where $x\in R,y\not\in R$ or $x\not\in R,y\in R$. A closure system $(X,C)$ is a $T_{1}$-closure system if $\{x\}\in C$ for each $x\in X$. It is easy to convince yourself that the specialization ordering on a closure system is a preordering and the specialization ordering is a partial ordering if and only if the closure system is a $T_{0}$-closure system. A closure system $(X,C)$ is $T_{1}$ if and only if $x\leq y\Rightarrow x=y$ in the specialization ordering. In essence, the specialization ordering is trivial for spaces and closure systems that satisfy any sort of separation axioms, but the specialization ordering is very interesting for spaces and closure systems without the $T_{1}$-separation axiom. For instance, the specialization ordering gives a one-to-one correspondence between the topological spaces on finite sets and finite partial orderings as was mentioned in the answers by Todd Trimble and Tom Goodwillie.

A based lattice is a pair $(L,A)$ such that $L$ is a complete lattice and where $L=\{\bigvee R|R\subseteq A\}$. If $(L,A),(M,B)$ are based lattices, then a based lattice homomorphism from $(L,A)$ to $(M,B)$ is a function $f:L\rightarrow M$ that preserves all least upper bounds (i.e. $\bigvee f[R]=f(\bigvee R)$ whenever $R\subseteq L$) and where $f(a)\in B$ whenever $a\in A$.

The category of based lattices is equivalent to the category of $T_{0}$-closure systems by the following functors. For notation, if $X$ is a poset and $A\subseteq X$, then let $\downarrow_{A}x=\{a\in A|a\leq x\}$, and let $\downarrow x=\{a\in X|a\leq x\}$. If $(L,A)$ is a based lattice, then $(A,\{\downarrow_{A}x|x\in L\})=\mathcal{H}(L,A)$ is a $T_{0}$-closure system and the specialization ordering on $A$ is the ordering inherited from the lattice $L$. If $(X,C)$ is a $T_{0}$-closure system, then $\mathcal{G}(X,C)=(C,\{\downarrow x|x\in X\})$ is a based lattice.

The topological closure systems have a few nice characterizations in terms of based lattices. If $L$ is a lattice with least element $0$, then we say that $a\in L\setminus\{0\}$ is join-prime if $a\leq a_{1}\vee a_{2}$ implies $a\leq a_{1}$ or $a\leq a_{2}$. We say that $a\in L\setminus\{0\}$ is join-irreducible if $a=a_{1}\vee a_{2}$ imples $a=a_{1}$ or $a=a_{2}$. Every join-prime element is join-irreducible, and in a distributive lattice, every join-irreducible element is join-prime.

$\mathbf{Proposition}$
Let $(L,A)$ be a based lattice. Then the following are equivalent.

$\mathcal{H}(L,A)$ is a topological closure system.

each $a\in A$ is join-prime in $L$

the lattice $L$ is distributive and each $a\in A$ is join-irreducible in $L$

the lattice $L$ is a coframe and each $a\in A$ is a join-irreducible in $L$. ///

Therefore, by the above proposition, one should think of topological spaces as based lattices $(L,A)$ satisfying the above conditions. In fact, from the above result one easily obtains the duality between sober spaces and spatial frames and other dualities. This correspondence between based lattices and closure systems seems to be the most useful when studying closure operators or topologies that are not $T_{1}$.

This duality between based lattices and closure systems is very interesting and I have found it to be very useful, but it does not seem to be very well known :(.

At last, I should mention that while we would intuitively want topological spaces to satisfy the Hausdorff separation axiom, I would not consider just any Hausdorff space to be geometrically pleasing. It is better to draw the line between the spaces that have a clear geometric or analytic picture and the spaces that do not at complete regularity. All completely regular spaces have a nice geometric picture since they can be embedded into a product $[0,1]^{I}$. The completely regular spaces are precisely the spaces that have a compatible uniform structure. Furthermore, the completely regular spaces are precisely the spaces that have a compatible proximity structure. One would expect a space with a clear intuitive geometric picture to be compatible with extra structure such as a uniformity, a proximity, or even a local proximity (local proximity spaces are completely regular as well). Furthermore, a majority of the interesting topological spaces tend to be completely regular. I have never in practice come across an interesting topological space that is Hausdorff space that is not completely regular other than a couple of counterexamples.
In fact, there are many classes of spaces that are automatically completely regular including the locally compact Hausdorff spaces, paracompact spaces, Hausdorff topological groups, uniform spaces and proximity spaces as mentioned before, metric spaces in particular, the order topology and even the lower limit topology on any ordered set, and even CW-complexes. The AMS mathematics subject classification puts the "lower separation axioms" (54D10) from $T_{0}$ to $T_{3}$ and starts the higher separation axioms (54D15) as complete regularity because there is a fundamental difference between spaces that are completely regular and the spaces that are not completely regular.