You could even follow the Unix model and do pretty well keeping the date and time (down to the second) in a single byte...
You are confusing bytes (8 bits) with words (32 bit or 64 bit depending on the system). A byte will only hold 256 discrete values so you could perhaps store the year in there but certainly not an entire date, let alone the time as well.
There was a time once when converting an integer value to and from ascii was prohibitively expensive (hard to imagine now), so it was not unheard of to store a year as 4 bytes.
APL hackers do it in the quad.