Sure.
Here's an example, a quad-core ARM chip from Freescale for £33: http://uk.farnell.com/jsp/search/dis...sp?SKU=2253171
It's got 624 pins. You'll need to design and press your own motherboard, and solder the chip to it.
Here's a corresponding 1GB RAM chip: http://uk.farnell.com/hynix-semicond...bga/dp/2147278
And 2GB of flash storage. You might want more than one of these. http://uk.farnell.com/hynix-semicond...sop/dp/1712425