How Would You Extract Data from Public Database?

There are two databases that I'd like to, for lack of a better explanation, download from the websites they are on. Both websites do not give or sell their databases in one big file, but do allow unlimited, free access to ALL of the data, but in an awkward manner (e.g., all names starting with an "A", etc.). Is there a way to "download" all of the data in an efficient way (rather than this "slow boat" method)? Are there companies that do this work for a reasonable price? I can do it myself but would love to save time if there are people who know the smarter, faster way to obtain this publicly accessible data. Thoughts or suggestions?