you could probably start 2000 plants in 40 nursery trays...each standard tray is about 3 sq. ft.
go figure....
and my measurements may be off but you get the idea of how to figure it out.
and in order to achieve germination i think you would need a heated structure.

A can fit 2000+ seedlings in my 12 x 24 greenhouse. It's probably much cooler here in the spring than there so I'm not help with heated vs unheated. I definitely have to heat mine.

I start mine in 72 cell plug trays and if I did the math correctly a bench 4'x12.5' would be enough space for 2000 plugs of this size. This size plug gives you a pretty nice size starter plant to transplant to a 4" pot or place directly into the ground. If you are going to transplant into pots you will need more bench space. But a 12x24 hoop house would be more than adequate.
Bottom heat will be a tremendous help and I find that it takes about a constant 90 degree soil temp to germinate peppers and I use around 80 degrees for tomatoes. Without bottom heat I would guess the tomatoes would finally come up but you might have trouble getting the peppers into the mood.
You'll need plenty of light and if you grow them on the cool side once they germinate you should get some nice stocky healthy looking plants. My guess is that you will surely need a heat source in the house but I don't know what kind of weather you have.
Bob

Thanks for all the answers. I may have overestimated my size needs.

Perhaps the wife would go for 2 homemade versions of these in a spare room.

Thanks,

Matt

Here is a link that might be useful: Light Cart

Nice looking light cart. I made a homemade version of these about 10 years ago (using 1x2's, but it only holds 6 trays so doesn't need to be all that strong: hinged at the top to fold flat when not in use) and it's still going strong. Mine has more headroom for each shelf than the one in the pic because I grow my tomato seedlings on in 4" pots on it until they are about 12" tall including the pot.

Kevin