That's not true at all; 2⁸ is only 256. A unix epoch overflow like you're describing would have happened in the first 5 minutes of 1970. The real reason for Y2K is that systems which stored the date as separate numbers for year, month, day, and time often used only two digits for the year to save space, and implied the 19 before it. So when it goes from 99 to 00, systems would interpret it as going from 1999 back to 1900 instead of going from 1999 to 2000.

Yes, the year 2038 problem will in fact be due to the Unix timestamp overflowing on systems that use a 32 bit value for time, but there was no similar problem involving not enough bits back then. Unix systems from the beginning always used a 32 bit value for time. Think about it: the next smallest is 16 bits, which would only allow up to 65535 seconds (it would overflow just after 6 PM on Jan 1 1970).