Like Windows 7, Windows 8 has Windows Easy Transfer, which will automatically transfer documents, email and settings from your old computer to your new one. So you can effect a transfer of this material using a transfer cable between the two computers, a network connection, or external media like USB drives or external hard drives. Read this article to see how it works. http://www.bleepingcomputer.com/tutorials/transfer-data-from-old-computer-to-windows-8/
You don't have to download the Easy Transfer program from their links; the new computer will put it on a thumb drive for you when you start the transfer program.
If you don't want all your settings to transfer and just want access to old files you certainly can install the old drive in the old computer or put it in an external enclosure and access the files that way. It's harder to find things like your address book, old emails, etc. that way, but it can be done if you know where to look.