There have been reports of people living exceptional long lives for thousands of years. This is extraordinary, but it. Is within the realm of possibility. There is a great deal we don't know about how people age and what will keep us alive and healthy for a long time. It is possible that this man stumbled on the perfect combination for extreme longevity.
While it's certainly possible that Gotho could be 145 years old, I won't believe it until I see more evidence. The oldest person ever recorded lived to be around 120, so I'm definitely skeptical. Average life expectancy at birth should continue to grow with societal and medical advances, but such a big increase would be surprising.
While I would not go as far as to suggest that documents and records held by the Indonesian government have been falsified I do find it dubious that these records have been kept accurately or are immune to mistakes or wishful thinking. There is some advantage to housing the world's oldest person.
The documentation that claims Mbah Gotho was born in 1870 is not real. A human cannot live to be 146 years old. There is simply no way that this age could b accurate. Yes, there is a slight possibility that Gotho might have been born in the late 1800s, but even that is a remote possibility.