This is one of the most difficult questions to answer for those living alone, or with aging relatives. My first bit of advice is to get a competent estate planning attorney to help you create an estate plan — preferably a Revocable Living Trust — and associated powers of attorney for healthcare, and a living will. If you develop a relationship with that attorney, perhaps he or she could serve as a co-trustee along with your youngest, most responsible relative.
If that’s impossible, you basically have two choices. Many banks will act as trustee for an estate — but their fees can be high, and if your estate doesn’t run into the millions, you won’t get their best attention. Recently, and in recognition of this need, there is a new designation: “professional trustee.” And they’ve created an organization to set standards and allow people to search for registered, and bonded, trustees. That would be a good place to start your search for an independent trustee. Their website is TrusteeAlliance.com.