How is professional history able to defend itself against postmodernism and the likes of E.H. Carr?

This is a strangely loaded sort of question. Is postmodernism attacking professional history? I seem to have not noticed it. Is E.H. Carr posing a threat to history? That is surprising to me, since pretty much every history grad student reads his work.

A more neutral approach would be, what is the historical community's take on postmodernist philosophy? And that can vary a lot, because it is a rather large community, and "postmodern philosophy" is notoriously wooly in definition.

There have been, for example, attempts to argue against specific "postmodernists." E.g., Derrida's Archive Fever is an attack, of sorts, on the idea of archival memory. This did provoke some responses, like Carolyn Steedman's "Something She Called a Fever: Michelet, Derrida, and Dust," in the American Historical Review, where she basically argues that historians have authority because they visit archives and steep in their glories. Or something along those lines. Other historians, like Dominick LaCapra, reject Steedman's approach to knowledge and archival authority as epistemologically naive. Some historians find some use in Derrida, many find no use from him. I don't think any regard him or his work as much of a threat anymore (more of an annoyance, at worst — there is nothing so tedious, if you are trying to actually understand the world, as someone who only wants to play language games).

This is just an example. In my experience most historians do not spend a lot of time thinking about deep epistemological problems of the sort that postmodernism tries to address. Historians of science do, but we're a strange bunch. I know a few historians of science who describe themselves as postmodernists, but they are mostly of an older generation. Most of the ones of my generation I would describe as neither modernists nor postmodernists — neither of these terms encompass the full range of possible epistemological stances.

There are several philosophers and thinkers who have been influential on some approaches to history that have at time been described as postmodernist (e.g. Derrida, Foucault, Deleuze), though if you talk to the philosophers they generally won't classify them as such (and many of the philosophers themselves denied these labels).

There was a time, in the mid-1990s, when there was this view that postmodernists posed some huge threat to science and university knowledge (the "Science Wars"). That seems pretty dead-letter by now, and more a symptom of things going on at the time (various "Culture Wars" coupled with a post-Cold War decrease in science funding in some areas) than a genuine philosophical struggle. The idea that postmodernists pose some sort of big threat to anyone strikes me as pretty amusing at this point. Actual postmodernists sequester themselves so thoroughly from the real world, and cloak themselves in so much jargon, that the chances of them actually influencing anyone or anything significant seems rather low.

On the other hand, though, some of the postmodernist destabilizations have had longer, more subtle effects on how we think about knowledge, politics, power, and so on. But it has not led to the radical rejections of authority that people worried about. There have been radical rejections of authority in recent decades — "we are sick of experts" — but I don't think you can plausibly blame that on movements in academic philosophy.