It's definitely possible. Has someone built exactly that? I don't know; there are hundreds of variants of single-board computers, and I'm only familiar with a few. It's not something that you'd build by buying a couple of parts and soldering them to a breadboard.
As for attaching memory, GPIO probably gives you a transfer rate on the order of a few MB/s. There are various things like SPI-connected SRAM chips, but those don't provide a lot of memory (think KB, not GB).
As for attaching memory, GPIO probably gives you a transfer rate on the order of a few MB/s. There are various things like SPI-connected SRAM chips, but those don't provide a lot of memory (think KB, not GB).