I had the random thought about whether British accents sounded different 300 years ago than they do today, and if they did if there is any way that we could possibly know that. Also, are there any interesting examples of figures of history that had interesting voices or speech impediments?
Unfortunately the answer given by /u/Ue-MistakeNot is not quite correct. While the linked video does show an alternate pronunciation, it's not actually the pronunciation of Shakespeare. If you see other videos of the same theatre troupe and following the same research, they readily admit that each actor has their own version of it, that there's not one form, and that it's kinda like Jurassic Park. By that I mean they used dinosaur DNA, but there's a lot of bullfrog in there too.
The actual answer to your question (tl;dr:) is "no, but we can reconstruct something close to it."
It's something that approximates, but we actually don't know that we've managed to get a perfectly accurate reconstruction. And that's what's been done in the linked video. They tried, and managed to get partway there, but then filled in the gaps (which again, they readily admit).
So how did they get close then? The most common way we reconstruct pronunciations is through what's called the comparative method. This is exactly what it sounds like. This is useful when we have a number of attested languages (usually modern spoken dialects/languages/varieties) that we know are related. From these we're able to compare them and determine how things sounded in the past.
Quoting Lyle Campbell's Historical Linguistics: An Introduction, "The aim of reconstruction by the comparative method is to recover as much as possible of the ancestor language (the proto-language) from a comparison of the descendant languages, and to determine what changes have taken place in the various languages that developed from the proto-language." This is done by taking the currently spoken ancestors of the parent languages and looking for correspondences. He goes on:
Currently existing languages which have relatives all have a history which classifies them into language families. By applying the comparative method to related languages, we can postulate what that common earlier ancestor was like - we can reconstruct that language. Thus, comparing English with its relatives, Dutch, Frisian, German, Danish, Swedish, Icelandic and so on, we attempt to understand what the protolanguage, in this case called 'Proto-Germanic', was like. Thus, English is, in effect, a much-changed 'dialect' of Proto-Germanic, having undergone successive linguistic changes to make it what it is today, a different language from Swedish and German and its other sisters, which underwent different changes of their own. Therefore, every proto-language was once a real language, regardless of whether we are successful at reconstructing it or not.
Where we see words that look similar in related languages and have the same meaning — English ship, Dutch schip, German Schiff, Danish skib — we can start to formulate what form that word would have originally had. While language does not always change in the same way every time, there are certain trends we can see again and again in language change througout the globe, so that changes we see in Swedish might also be happening in Shanghainese (e.g. /a/ tends toward /o/ which tends to become /u/ whi.ch tends toward /y/, a process called "back vowel raising" and identified by Labov for Swedish but which also shows up perfectly in Shanghainese). So we can look at these modern words for "ship" and figure out that the original word was something like skip. Realise that I'm simplifying here.
This is fine for languages that weren't written down, or which were written in a way where pronunciation isn't really clear (early Chinese). For things like Latin and Sanskrit, it's a bit easier, because these alphabets have survived into the present in various forms.
Yes! I'll use Shakespeare as an example because I know of a video that demonstrates the difference in pronunciation. It sounds very close to a Devon accent, but with some differences.
2:52 For one example of the difference.
3:40 for their explanations of how they know what it sounds like. I'll summarise them below. (Unsure if that's within the rules or not, please let me know if more needs to be added).
We have a few ways of knowing.
One is rhymes. Shakespeare uses some rhymes that don't seem to flow or make much sense if we use modern pronunciation, but if we use OP (original pronuciation), then it starts to fit. This giuves us a clue as to how certain words sounded. Proved and loved were rhymes in OP, because proved was pronounced 'pruv-d', as one example.
Related to this are some puns that are included (the video mention one to do with lions and lines, which would have been pronounced very similarly) in his works, and that only start making sense when you pronounce them certain ways (the same way that puns in text dont work well when spoken). 4:52 in the video for some examples where you can hear the difference.
A second bit of evidence is how people from the time commented on how things sounded. This also ties in to the rhymes, because some words were noted to have rhymed. The video mentions the pronunciation of 'r' in a certain place being commented on as being a 'doggy sound' like a 'grrr'. Things like that give us clues.
The third bit of evidence is spellings. At that time spelling was more closely correlated to the actual sound of the word. From these you can get a lot of distinction in how it was to be pronounced. This might be the least powerful piece of evidence, but it does provide another way to puzzle out how to pronounce it if the other way wont work.