These examples seem to be very difficult to construct. The problem is that any local compactness or uniformity will automatically boost your space to a Tychonoff space, and Tychonoff spaces are closed under passing to subspaces or products. Consequently, there's doesn't seem to be a "machine" for producing these kinds of spaces.

The idea of all the counterexamples $X$ is to write down enough open sets of $X$ to make it clear that points can be separated from closed subsets, but to somehow rig things so that any continuous real-valued function on $X$ identifies two distinct points of the space.

The example in Munkres's textbook that Elencwajg mentions is a pretty straightforward one (relatively speaking); it's the same in spirit as Raha's example, which is the easiest I've found. Here it is:

For every even integer $n$, set $T_n:=\{n\}\times(-1,1)$, and let $X_1=\bigcup_{n\textrm{ even}}T_n$. Now let $(t_k)_{k\geq 1}$ be an increasing sequence of positive real numbers converging to $1$.

For every odd integer $n$, set $$T_n:=\bigcup_{k\geq 1}\{(x,y)\in\mathbf{R}^2\ |\ (x-n)^2+y^2=t_k^2\}$$ and let $X_2=\bigcup_{n\textrm{ odd}}T_n$. Now let $$X=\{a,b\}\cup\bigcup_{n\in\mathbf{Z}}T_n$$

Topologize $X$ so that:

- every point of $X_2$ except the points $(n,t_k)$ are isolated;
- a neighborhood of $(n,t_k)$ consists of all but finitely many elements of $\{(x,y)\in\mathbf{R}^2\ |\ (x-n)^2+y^2=t_k^2\}$;
- a neighborhood of a point $(n,y)\in X_1$ consists of all but a finite number of points of $\{(z,y)\ |\ n-1<z<n+1\}\cap(T_{n-1}\cup T_n)$;
- a neighborhood of $a$ is a set $U_c$ containing $a$ and all points of $X_1\cup X_2$ with $x$-coordinate greater than a number $c$;
- a neighborhood of $b$ is a set $V_d$ containing $b$ and all points of $X_1\cup X_2$ with $x$-coordinate less than a number $d$.

This is a space that is $T_3$, but every continuous map $f:X\to\mathbf{R}$ has the property that $f(a)=f(b)$, so it is not $T_{3\frac{1}{2}}$.