I'll echo a bit of what has already been said but with a few differences from what I've experienced. Xbox emulation is a complete no go currently, problem being with how close the architecture was to a traditional PC, no one has been able to emulate the platform consistently. However, Gamecube/Wii, PS2, and Dreamcast are very doable and actually improve upon the experience with the right hardware. Saturn emulation has also gotten much much better over the past few years and most things are emulated quite well at this point.
To get an idea of what you might need specs wise, it may be helpful to look at the emulator's documentation to see what they recommend:
https://dolphin-emu.org/docs/faq/ - Gamecube/Wii emulation
http://wiki.pcsx2.net/index.php/PCSX2 - PS2 emulation
If you're meeting or exceeding the specs listed on these pages, you should be good to go for everything previous to these systems. The onboard graphics card in those is really what will bottleneck your performance. A dedicated GPU is somewhat essential for good emulation performance on these relatively new systems. To be honest the newer Celerons and Pentiums aren't too bad from a CPU standpoint, they are just cut down versions of the higher end processors. For single or dual threaded tasks they tend to keep up with their big brothers. Keep in mind though generally a faster dual core is better than a slower quad core when it comes to emulation.
Really what it comes down to is what you want out of the experience. If you want to run PS2 games at 1080p with improved texture packs and anti-aliasing, what you have listed wouldn't really be up to the task. At best, you'll be getting a base experience with some frame drops due to the weak GPU. Have you ever thought about building your own? Likely you could get something better than what's offered here and it would be expandable down the line if you wanted to get into PC gaming.