Tim Henderson who blogs at Earliest Christianity asks, “Can historical Jesus work be done by those who affirm inerrancy?” I can think of a few people who I think affirm inerrancy and do well in the arena of historical Jesus studies: Nicholas Perrin, Michael Licona, Craig Blomberg, and Darrell L. Bock come to mind (Ben Witherington?). Most conservatives that are well-respected do not: Tom Wright, Scot McKnight, Larry Hurtado, and Richard Bauckham are a few examples.
When I have read people like Perrin and Licona they seem to do well by honestly assessing whether or not something seems likely on the basis of historiographical regulations. I would guess that if they were pressed on whether or not “corps came from the grave after Jesus’ resurrection” they would say something like it is historically unlikely but they believe that it did happen based on their confessional stance toward Scripture. This does not mean they think it is historically verifiable or even likely (e.g. the resurrection, in some sense, is not historically “likely” since dead people stay dead), but rather that their theological convictions lead them to affirm it’s truthfulness on different grounds.
So yes, in some sense, one can affirm inerrancy and do historical work as long as one is willing to admit they believe something happened on the basis of a confessional stance rather than on its likelihood when examined through the lens of a historian.
Now whether or not a scholar who affirms inerrancy will have broad influence in his/her guild is a different story. That person must decide whether or not their confession is something that they (A) hold deeply; (B) will willingly allow to impact the breadth of their audience; and (C) will allow to determine whether or not they must teach in institutions where this confession is necessary. If any of these three things is not acceptable then said scholar better seriously consider asking themselves hard questions about their understanding of the doctrine of inerrancy.
So my yes and no answer is this: Yes, you can do historical work while maintaining inerrancy though you must wear two hats (one of a historian, one of a Christian with a particular confession) and be willing to believe there may not be good historical support for something you affirm as true on different grounds. No, you cannot do it and gain as broad an audience as someone who appears more objective. I emphasize “appears” because I don’t think an atheist or a Liberal Protestant (or an atheistic Liberal Protestant) is more “objective”, per se.
Anyways, that being said, go to Tim Henderson’s original post to interact with what he asked and do feel free to comment here and let me know what you think of my response.