What's the name for "negative length" in pseudometric space? In a pseudometric space (a metric space with signature $(1,1)$, for example), the "distance" between two points can be negative. That happens (in this example) when $d^2 = x^2 - y^2$ is negative because $y > x$.
But people don't like when I talk about "negative length" because apparently, the words "length" and "distance" are defined as always being positive.
In the case of 4-dimensional spacetime (which is Riemannian), the word to use is "interval."  But what's the word to use for negative distance in the general case of any pseudometric space?
 A: In your example $d^2$ is unfortunate common notation (since d doesn't exist as a real number when $d^2$ is negative ) for what you should call a quadratic form which could be used  for 1 dimensional space time where x (say in feet) is the space coordinate  and y (say in seconds)is the time coordinate . It is not a pseudometric in the (x,y) plane .the metric used is the usual one D((x,y),(x',y')) = $((x-x')^2+(y-y')^2)^{1\over2}$  which is the 2 dimensional euclidean metric . 
In general a pseudometric D on a set M satisfies D(a,b)=0 ,D(a,b)$\ge$0 ,symmetry and the triangle inequality where a,b are in M ;but leaves out that D(a,b)=o implies a=b . 
A: 
[...] the "distance" between two points can be negative. That happens (in this example) when $d^2 = x^2 − y^2$ is negative because $y \gt x$.

If $d$ is supposed to denote the distance value under consideration then the example you gave is not applicable because: the square root of a negative real number, i.e. for all cases $x^2 - y^2 \lt 0$ is not a negative real number but of course imaginary.
(And whether there may be objections to calling and considering values of imaginary or of complex numbers as values of "distance" or of "length" is a separate issue.)
An applicable example, somewhat based on the above, is instead:
$$d : \mathbb R^2 \rightarrow \mathbb R, \qquad d \mapsto \text{Sgn}[ \, x^2 - y^2 \, ] \, \sqrt{ \text{Sgn}[ \, x^2 - y^2 \, ] \, (x^2 - y^2) },$$
where $\text{Sgn}$ denotes the sign function.

But people don't like when I talk about "negative length" because apparently, the words "length" and "distance" are defined as always being positive.

Clearly, the (applicable) example function $d$ above does not satisfy all defining properties of a (plain) metric space.
Still, at least it shares one property of (plain) metric spaces, namely "Indiscernability of the Identical": $\forall x : d[ \, x, x \, ] = 0$.
(This illustrates by the way that even in the plain notion of distance its values are not always defined as strictly positive.)
However, obviously there are generalizations of the "plain" notion of distance.

In a pseudometric space

Example function $d$ defined above is (obviously) not an instance of pseudometric distance either. Therefore the relevant "larger" questions would be, to which type of generalization belongs the above example function, or how to call such a generalization of "plain" metric space.
For lack of knowing of any suitable precedence (on this website, in Wikipedia, or elsewhere), I have already proposed here to call the particular type of generalization examplified by the function $d$ above as quasihypometric distance.

In the case of 4-dimensional spacetime (which is Riemannian),

... actually, pseudo-Riemannian and even more specifically Lorentzian ...

the word to use is "interval."

Yes values of spacetime interval, $s^2$ can have negative real values (as well as non-negative values). But (alledgedly), the notion of spacetime interval is not applicable to Lorentzian manifolds in general, but only to flat manifolds (such as Minkowski space). A more general notion would be Synge's world function, $\sigma$.

But what's the word to use for negative distance in the general case

A notion for referring to the separation between timelike (or lightlike) separated events in general is Lorentzian distance.
(Apparently, terminology for naming the quantities $\text{Sgn}[ \, \sigma \, ] \, \sqrt{ \text{Sgn}[ \, \sigma \, ] \, \sigma }$ still needs to be proposed.)
