On investigation into this issue, it appears that in our effort to cache certain items of the site to keep things fast, our cache store ran out of room and as a result places in which the cache was being used would begin to throw errors, these in turn quickly filled the error logs.
With the cache filling up, and the error logs being written to at a high rate, the server soon filled up, and compounded to the problem.
We've fixed the issue as it currently stands, and will be looking into better systems to ensure that this event doesn't take place in future, or is at least better handled.