You're gonna wanna get your ps2 to the point where it can boot elf (ps2-executable programs) files, how you go about doing this i'll leave to the other forum users to sway your opinion and yourself to choose, but there's the hard mod method of installing a mod-chip which I have no experience with, and there's the soft-mod method of running the memory card exploit via a program such as HDLoader.
Programs you're definitely going to need are a launcher (Such as Ulaunchelf) and, of course, snes-station.
What you're looking for is completely possible, you can have the playstation completely enclosed save for the power button and have this all set up running fine.
(What I'm about to explain would be how a soft-modded ps2 would work)
You'll only have one disc to ever need to be in the ps2 and that's going to be an original psx trigger disc, whenever you turn the ps2 on it's going to boot this disc and find a file on the memory card that tells the ps2 to boot a certain file off the memory card instead of booting the psx disc. This file would be something like ulaunchelf which is a program designed to launch ELFS. Ulaunchelf, in turn, would be set up to automatically boot the SNES-Station elf, and from there you'd be brought to the snes-station program and presented with your list of snes games to play =)
Another good emulator is PGEN which is a Sega Genesis emulator. It runs roms off the hdd as well and it is imo just as reliable as snes-station
Hope this helped
Ver. 3 NTSC independence exploited ps2, 120 gig WD HDD, HDLoader .8c, Ulaunchelf 4.04, Blackrhino linux on WD 40 gig HDD