Where would the money come from?
Since it's impossible to know whether you'll even have these types of expenses, you may not want or be able to save enough for worst-case scenarios.
That's understandable, but you should have a plan for how you'd deal with them if necessary.
Here are some common options:
Your home. If you have a permanent need for long-term care, you may not need a separate dwelling anymore. If your home is paid off, you can plan to sell it and use the equity if needed. (This could also free you from real estate taxes, utilities, and other housing-related costs.)
An income annuity. These annuities won't pay for long-term care specifically, but they'll guarantee you don't run out of money, even if you spend all your remaining savings.
Long-term care insurance. If you buy a long-term care policy, it will pay for some of these costs. Keep in mind there are often waiting periods for the insurance to kick in as well as maximum payouts. And once you're older than your mid-sixties (or if you're not in good health), premiums for a new policy are usually too expensive to be worth it.
Medicaid. The government health insurance program that covers low-income Americans will pay for long-term care, but generally only after you've exhausted all your assets.