For sure in about 4-6 billion years time when the fuel powering the sun (mostly hydrogen) runs out – before that it will get hot. As humans we’ve been around for 200,000 years and been quite inventive so we have 4 billion to get our act together and work out how to get to another planet surrounding another sun-like star… it will have to involve some clever tunneling through space-time since the closest amenable stars are millions of years away (travelling at the speed of light – which we can’t…!).
On a shorter timescale I think the issue of population growth and dwindling natural resources is far more serious and that could see us extinct well before the sun goes out… but again that is something that is partially under out control..
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