No, gross domestic product is not the best indicator to measure standard of living.
There are a number of problems with GDP as a measure of standard of living. First, GDP does not measure how much income an average household gets. China has a larger GDP than Japan, for instance. But it has so many more people that each household gets much less than each Japanese household. For this reason, GDP per capita would be a better measure.
Beyond that, GDP in any form does not capture issues of quality of life. A country whose people work all the time and never have any leisure might have a very high GDP per capita. But do its people really have a better standard of living and quality of life than those who make less money and have more leisure? GDP cannot measure this. Of course, it is very hard to measure happiness and quality of life in an accurate way.
Finally, GDP does not measure the distribution of income. The GINI coefficient is much better at this. GDP does not tell us if a few people have all the wealth and a high standard of living while the majority of people are poor.