How difficult is it to study a period of history in depth without understanding the language of the time? For example, I imagine studying Roman history would be extremely difficult without an understanding of Latin, but is this also true for studying Anglo-Saxon history without knowledge of Old English? Also how much can modern language understanding help in ancient study, say studying Celtic history and only being able to speak modern Gaelic.
Do you mean historical linguistics or knowing the languages? Historical linguistics is the study of how languages change over time, it's what's used to reconstruct things like Proto-Indo-European. It's not the same as knowing the languages, a historical linguist doesn't necessarily actually know the language that he's working on, although for obvious reasons it helps. It's also generally not all that helpful for history, although it can be useful for learning the languages (I don't personally think you can learn Greek without some basic idea of how the Greek language changed from prehistory to Attic, because otherwise you have to memorize the paradigms of literally every verb you encounter like a psycho). Knowing the languages, though, is of great use. I would argue that it's nearly impossible to study ancient history and classics without knowing Greek and Latin (although there are a very few number of scholars who actually don't). In more contemporary fields maybe it's not as important, I don't know--as a classicist I deal more or less exclusively with the texts themselves, so I would be forced to work from translation, which is very unsatisfactory for any detail or nuance and is not always possible, as some texts have never been translated. What's important in any historical field is being able to read the sources, whether that's direct material or scholarly material