I'd like to start my seedlings really early next spring, but I don't have
much room indoors. I'm going to set up a wood-frame and plastic tarp
enclosure outside. I realize I'll have to get the seedlings to sprout
indoors, and then I'll move the trays out to the enclosure asap.

Any advice on this? How early can I start this? Will the seedlings continue
to grow when it's really still quite cold outside?

I'm in s/w BC (zone 8). Pacific Northwest for Americans. The temp only
occasionally goes below freezing in winter, generally hovers in the 40's.
Comments?

I think you are in a good situation, relatively speaking. Seedlings
actually prefer cooler temperatures (in the 60s) once they have
germinated. Start them on top of your fridge, where it is warm, then
move them outdoors. Find a way to moderate temperature swings inside
the enclosure. You can have it under ground, you can add water
containers (I have a 55 gallons drum), and you can have ventilation.
Sterilize your trays well because the conditions will be generally
conducive to damping off.