@itinere,
If you've got enough discretionary income, see if you can put them up in a lower budget hotel or long-term temp rental (vet it first of course, maybe with your sister) for a while. I recognize there are parts of the US where "cheap" means $150+ per night (I live in one of them).
But some long-term temporary housing providers might offer a monthly rate or the like. I just Googled long term temporary housing and got some hits, including this site, which seems to have good info:
https://www.extraspace.com/blog/moving/moving-guides-tips/best-websites-finding-short-term-housing/
If there is a college or university nearby, they may also have month to month rental space for grad students. If they need to fill a slot,
and if they take kids (they might not), that could be another place to look.
Make it excruciatingly clear to your sister and her husband that this is a loan and not a gift. You expect to be paid back within 6 months of them finding their own place. And you also won't pay for this past a year. Once the 12 months are up, they are on their own.
But quietly and in private (that is, don't tell them), write it off. You're highly unlikely to get any of it back. The idea behind the loan vs. gift speech is to get them to act quickly. Part of their acting quickly should include your sister looking for at least temporary employment and inexpensive daycare. Her being a full-time SAHM is a luxury that she and her family just plain can't afford right now.
And, not for nothin', but they are guests in your home. If their kids are breaking your stuff, then your sister needs to get her act in gear and get them to call the hell down. That is her job right now and she needs to be doing it.
Will this be expensive? You better believe it.
Will it get them out of your hair? Yes.
Will it keep them from a sketchy shelter? For a year, yes.
Will it get them to move faster to get themselves situated? Probably.
Will it absolve you of guilt or bad feeling? I would say so.
Telling them to take a hike less than a month after they've lost everything is just plain cruel, no matter how godawfully annoying they are. But you don't have to be their port in a storm forever.
Oh, and another thing. Depending on their credit rating, they may be able to take out a loan to cover replacing the husband's specialty computer equipment (act quickly, as the Fed is about to raise rates). Or they can try crowdfunding. And if he can get his computer rig back, the husband might want to work with his long-term clients to see if he can get an advance on some payments, or take on some smaller jobs that he can finish quickly so they can have some cash on hand.
There may also be some loans or grants available through your local government or even organizations like the Kiwanis club. No one will ever know unless they investigate this and ask. Your sister is at home so she can do a lot of this legwork. And, in fact, she
should be doing this. Not just investigating grants, but also hunting for housing solutions.
Your sister and your BIL should make moving out of your home Job 1.
And they should buy some damned renters insurance. Maybe buy them a first month's premium as a housewarming gift (I'm not being snarky here; well, no more than usual--this is something they actually need).