The hard drive needs to be powered either by plugging into the wall or using a powered USB hub. There is a PiDrive hard drive that is low powered that might work. I believe it shares the power adapter with the Raspberry Pi. I also see they have an enclosure for the both the hard drive and the Pi in one case.

Another option instead of a hard drive would be to use a large USB flash drive:

I've never tried it but I've read that powered USB hubs can power the Raspberry Pi but I'm not sure if it will be the correct 5V 2.5A (or whatever the Pi 3 needs).

With the correct power adapter to your Pi, you can run the four PS3 controllers via Bluetooth without any issues.

then i can put in it pretty much complete romsets of games (from the NES to the Dreamcast)

I really haven't had many issues with NES, SNES, Gameboy (Color/Advance), Genesis, Sega CD & 32X, Game Gear, TurboGrafx 16/CD but it has been pretty much hit or miss with Dreamcast games. You might check out the Dreamcast Compatibility List:

The games that don't work, you really don't need to have because they are just wasted space (unless in the future there is an update to the emulator that supports that game).

PSX games are more reliable than Dreamcast but I've had issues with some of them as well. I don't know if the PSX has a "Compatibility List" like the Dreamcast.

P.S. making a wood box to 'store' all the components would be a super bad idea, right?

Yeah. Someone on this forum (maybe @obsidianspider) is making a box with arcade buttons & a joystick on top and all the hardware inside and then you just plug it into the TV. Gives you that arcade feel but the box just sits on a coffee table and you get to use your large HDTV! You could even make the box wider and add a 2nd set of arcade buttons/joystick for player 2. The player 3 and 4 can use Bluetooth PS3 controllers.