Yes a nursing home can legally take all your money as well as any other assets you own such as a home etc if you own the home.
Nursing homes are very expensive so you need to have some savings in place or some health insurance that will help cover the cost of nursing home care when and if you should need it.
Staying at home is usually much cheaper than staying a nursing home and you can have a home health nurse come by and take care of the you in your own home.
That way you can keep your home until you die unless someone else such as a spouse etc jointly owns the home.
If you owe a mortgage on the home then the bank will be the only one able to take the home and not the nursing home.
But they can take your money to pay for the nursing home.
You need to get a lawyer for nursing homes to help you plan ahead and to make sure you don't lose all your money in the even you need a nursing home and cannot pay directly for it.