I think he was just being playful and tossed out a number.
My maths show that 231 (we don't use the last but, it's for sign sigh...) /60 seconds /60 minutes /24 hours / 365 days equals 68 years. 1970 is the beginning of the epoch, so 1970+68=2038, the year the clocks overflow.
263/60/60/24/365 = 292,471,208,677 years. That plus 1970 means we can go until the year 292,471,210,647 (approximately) until we run out of seconds in a 64 bit int, if we don't use the sign bit. Twice that if we do.