As @whuber notes in a comment on another answer, there is no reason to restrict hazard functions to distributions having only non-negative support. Although a Wikipedia page suggests that restriction, a survival function $S(x)$ can be taken in general to be the complement of a corresponding cumulative distribution $F(x)$, with $S(x)=1-F(x)$. For any value of $x$ for which $f(x)$ is defined and $S(x)$ isn't 0, there is a defined value of the hazard $h(x)=f(x)/S(x)$.
A survival function defined over the entire real line can be useful in evaluating a parametric survival model fit, as explained for example in Chapters 18 and 19 of Frank Harrell's Regression Modeling Strategies. A Kaplan-Meier survival plot of censored, standardized residuals over the real line can help evaluate whether the distribution of residuals matches that expected for a particular choice of parametric family (e.g., standard minimum extreme value, defined over the entire real line, for a Weibull model).
Furthermore, survival analysis with left-censored survival times can be thought of as allowing for negative survival times. In R, a left-censored observation at time $x$ can be coded as Surv(-Inf, x, event).
Survival functions defined over the entire real line thus can be useful.
The question, however, is how useful a hazard function is for distributions with support over the entire real line. The hazard function is perhaps most useful as describing the probability of an event at time $x$ given that there hasn't yet been an event. In practice for analysis of event times, it's useful to define a time origin such that $f(x)=0$ (and thus $F(x)=0$) for $x \le 0$. In that case, $h(x)=0$ for $x \le 0$. I suppose there might be circumstances in which a different time origin might make sense, but it's hard to think of one.
