Modern technology can track people through a variety of ways. Cars now are able to be tracked unless so old because of bluetooth technology, GPS, etc. Cell phones (and even land lines) are tappable. The US postal service takes a picture of literally every piece of mail that goes through its system, and unless one could devise a way to let your friends and family know what's going on through different mail systems (maybe sending a letter from 10 different mailboxes in different zip codes vs. just 1). E-Mail systems, social networks, etc are hackable, thus a determined tracker could see what emails were sent. Those tracking them could contact their loved ones (via public records, social media, etc.) People's movements are monitored through cameras, so even public transportation wouldn't be safe from experts determined to get your location, unless elaborate disguises are used. Even wilderness areas aren't truly safe, as many are crisscrossed with roads or public access points - not to mention you could run into other people as well. If the authorities are intent on finding you, they'll use helicopters to search the wilderness.
This question hinges on people being able to find your location in ANY way: traditional, nontraditional, technological, social, etc. If they find you, you're caught. This must be an instantaneous and permanent disappearance, not one staged over time. Also wealth should not be considered part of the equation. Budget:what an average middle income family gets in a single paycheck to do this once and for all.
This is an "average Joe" no special training, equipment, knowledge, etc involved, but the ability to look up and learn stuff as they need to know it (for instance YouTube at the local library). Think the TV series/ movie "the fugitive" but with more modern technology and everything the government can throw at the one they're trying to catch.
Also, as a note to the answer that was given for the military person: people will still know this person exists. No records will be wiped. His friends and family will assume s/he is still alive, unless given proof otherwise. No photographs will be altered. No documents of his/her former existence will be altered in any way shape or form. S/He simply wants to cease to exist from THAT moment forward, not retroactively erase his/ her existence.