I appear to have set up my fortress in an area where there isn't any stone. So I can't build a forge or anything, because I have no flame resistant building materials. Am I screwed? Or can I get out of this mess somehow?
It depends on what version of Dwarf Fortress your running, but most likely, if you dig deep enough you'll find stone. If you are blocked by an aquifer there will be rock basically directly below it. If you are playing the latest version and there honest to goodness is no rock, you always have a source of magma. Dump water on lava, get obsidian.
You can always get around a shortage of anything by trading carefully. However, as I have experienced without any metal, certain things do get difficult with a shortage. Just try to hold out (by trading for example) as best you can, the worst that can happen is Fun.