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Google and Yahoo were founded shortly after the invention of the public internet. Before this (1960 to 1980, i think) there was at first the ARPANET (a military test network) and then a network between the universities for faster exchange. The shutdown of the ARPANET 1990 is seen as the start of the internet as we know it today. And there was nothing much mor than the "websites" of universities or computer companies. And via telnet there was the first "messenger" :D
First of all, be aware that "the internet" and "the www" aren't the same thing. The WWW is just a part of the internet, and it's this area that the likes of Google and Yahoo! focus upon. Other areas include FTP, email, torrents, gopher, and many more. The internet was much the same as it is now, though on a more basic level. Other search engines existed before Google and Yahoo! Yahoo! was the first of those two and originally involved a couple of students in a room at university manually looking at, categorising and rating sites. Back then, this was possible as there were only a couple of thousand sites. You found sites by starting at one place and following links to others. Trading links was common, as were pages with "useful links to other sites". Essentially, links going all over to create the "web" we use today. Sites such as Alta Vista and WebCrawler began to use automated tools (or "bots") to trawl this web and snatch bits of text to be used in the early search engines. Prior to this, you were lucky if an individual site had a search index to find information on its own pages! These days we simply couldn't do without the likes of Google and Bing to help us locate resources. So, in essence, the internet hasn't changed as such. It's just grown much bigger and we need these tools to help us manage it.
If I were to put my own website up on the web, would google be able to crawl to it and put it in its results?
Yes, unless you added a robots.txt that restricted it's access.