If a person has no family, what happens to them and who helps them out?

If there is no family and the person has not appointed anyone for helping it is possible that at some point the legal system steps in and they are appointed a Guardian. (these are often people that neighbors report to APS, or local departments find out like the Health Department, Fire Department or Police)
A person can appoint anyone they trust to be POA for Health, for Finances in most cases this would be a trusted friend or if they have no one it can be a lawyer that has drawn up POA.
This is why it is important to sit down with family and discuss what it is you want. Discuss where you keep important papers. Be open and honest. Appoint someone to be your voice when you have none.
Not an easy discussion to have but necessary.
Oh, an to make your son's lives easier...
Start getting rid of stuff now.
No one wants your cookie jar collection so sell, donate what you really don't want.
Make copies of family recipes so that they have them
Get rid of the clothes that are in the box in the attic.

Generally if one is competent to do so they put the future in the hands of a Professional Fiduciary, who will take over conservatorship as needed for a fee. If there are no funds whatsoever then a person often, upon need, becomes a ward of the State and a Fiduciary is appointed by the courts for guardianship, placement, management of funds, of application for State and Federal help when there are no funds.

For the people who actually plan *realistically* they must find someone else to fill the legal roles of DPoA and executor of their Last Will. There are actual professionals that can be hired to do this. For those who do not plan or don't have the financial means to hire a professional, there is usually some sort of medical crisis that gets them onto the radar of social services, at which point if no one else steps up, they move to secure guardianship of that vulnerable elder. At that point the elder gets a guardian who manages their everything: medical and financial, decides where they are placed, and their comfort. Then takes care of their cremation and disposes of their remains (I'm not sure where).

In regards to your sons, this is a very important discussion to have with them sooner rather than later. They need to be asked if they are willing to be involved in their parents' care, not "assumed" into this responsibility. You may be trying to carry over a cultural value but your sons are another generation removed from that and may not share it. As you all age this discussion should be had every few years or when there's a significant change in health, finances, etc. or even if you don't live near each other. FYI it is nearly impossible for adult children or family to know what exactly is involved in caregiving, so they often agree to something that, if they only knew fully what it actually involved (and often the physical, mentl, emotional, financial cost), would never agree to do. Kudos to you for thinking ahead and being realistic about your sons' involvement.