You don't directly inherit the debt, the estate does, and you get to keep whatever is left over. If Cole wants to claim his half, it will be similarly reduced. This is effectively inheriting debt in that the debt has to be paid off.
If the estate is underwater, then you can just not inherit anything and the debt goes away.
If the debt can only be paid off with one large, indivisible item in the estate (usually a house), and you want to keep that item, then you have to negotiate with creditors (and potentially other inheritors) for a way for you to inherit the debt too under a payment plan.
To satisfy the debt to all other parties, you can either refinance that item (in the case of a house, a new mortgage that lets you convert equity into money to pay the debt). There is enough equity to pay off Barb's debts, but Chris trying to get a new mortgage would be an uphill battle at best. If Chris can persuade the unsecured creditors to let him take on the debt (also an uphill battle), and Cole is uninterested, then no refinancing is needed.
If Barb's debt is in a joint account with Chris, then the debt was always Chris' too so it doesn't even go into the estate. Similarly, if Chris were a cosigner on Barb's debt then it automatically transfers to him, with Chris effectively inheriting it.
Cole will not inherit any debt unless he decides to (not bloody likely), but he can go to the back of the line to claim any remaining part of the estate, effectively becoming a debt for Chris if Chris wants to keep the house.