The Tesla CEO, worth a reported $68 billion, also criticised US lockdown measures, which some experts claim could have saved thousands of lives.
He said the pandemic has “questioned my faith in humanity” because people have become “irrational”.
President Donald Trump has been pushing for a vaccine to be rolled ahead of November's election. But with the leading candidates still in trial stages, a shot may not be ready until spring/summer 2021.
Even then, there is no guarantee that any vaccine will prove to be effective in tackling the disease, which has claimed some 1 million lives globally.
Speaking to the journalist Kara Swisher for an episode of the New York Times podcast "Sway", Mr Musk, 49, said he doesn't plan to get a vaccine because he and his children are "not at risk" from coronavirus.
It was not clear what Mr Musk, who recently had a baby boy with wife Grimes, 32, meant when he said he and his family are not at risk from coronavirus.
However, data suggests morbidity rates are much higher among the elderly and those with underlying health conditions, such as asthma and obesity.
During the NYT interview, published on Monday, Mr Musk, who has been outspoken about the pandemic, also hit out at blanket lockdown restrictions.
"Essentially, the right thing to do would be to not have done a lockdown for the whole country but to have, I think, anyone who's at risk should be quarantined until the storm passes," he said.
Mr Musk has made similar comments in the past. In a separate interview with the NYT in July, he downplayed the risks for children and young people, who are not immune and can still get very unwell.
The South Africa-born entreprenure is among a growing number of Americans who say they will not get vaccined when a candidate gets approve by health authorities.
Some 30 per cent of respondents to a recent Ipsos MORI survey said the don't plan to get a vaccine when it becomes available.
Around a fifth of those people said they were opposed to vaccines in general, according to the poll.