You could use USBAdvance to "store and play" your PS2 games, although the compatibility rate is rather limited as to which ones it can play. But as far as storing & playing the PS1 games, I know of nothing that can do that, not even on a modded console.
For PS1 games, you can store it on a memory stick and use a modded PSP to boot it. You'll of course have to first convert the PS1 game to a format where the PSP will know how to read it. For PS2, the slim's are cool, but the original phat model has more potential and benifit. Use a hacked version of HDLoader along with an IDE HDD w/ network adaptor to store and play games. You can use USBeXtreme, but data transfers VERY SLOWWWWWW. In my opinion, I would consider the USB method, the very last resort.
