They would potentially legally be liable if they were involved in some way in your business or in the violation, act, infringement, etc. which caused the lawsuit. That means that if they let you live there for free (providing you the office space for free), let you use their internet, or otherwise provide the resources you used to do something wrongful, they could possibly be liable--certainly, even if in the end, they could defend themselves by showing that they are not in any real way involved, they could very plausibly be sued and forced to defend themselves against the lawsuit. You want to establish an "arms length" relationship and provide as much of your own resources as possible, to insulate them: e.g. provide/pay for your own internet and phone; use your own computer, which you can show is only yours and not a family computer; maybe pay them rent (with a written lease) for the portion of the house you use for the business (even if only $100 - $200 per month for a corner of a room and a desk); etc. (And if your business can't support such minor costs, it's probably not worth pursuing.)