It always amuses me how much people freak out over venomous spider bites. Unless you have a weakened immune system, are very young, are very old, or are explicitly allergic to the venom in question, almost all types of spider venom are fairly harmless on the whole. Even with the brown recluse, it's the exception to the rule that their bites cause effects like necrosis. The vast majority of brown recluse bites don't have any side-effects at all, in fact. Even the legendary black widow isn't half as deadly as people make it out to be (only the females are dangerous, incidentally; the males' bites just hurt, but lack any venom), and its venom is strong enough to warrant its own syndrome name. The effects of said syndrome being about on par with your average flu's symptoms and are overcome naturally about as fast (ie. a week or two for healthy individuals). The entertainment industry really has given us entirely the wrong impression about spiders.
Now getting bitten by a whole nest of black widow females... That would be genuinely dangerous no matter how good your immune system is. Other than that, though, your Dwarves shouldn't be bedridden for eternity with all of their insides rotting out from a single brown recluse bite.