If you want to store only the year, why not store the years since 1900? That way in a single byte you can go all the way up until 2155. If you want to store it bit-aligned rather than byte-aligned why limit yourself to 00-99? 7 bits can store the year from 1900-2027.
If you want to store day + month + year with a two digit year you have 365.25*100=36525~ options. A two byte value can store up until 65535 options, which can hold all days until 2079.
If you are storing your dates as flat strings (DD-MM-YY or so) then you are already wasting such a large amount of bytes that "storage concerns" seem moot. You are already using 8 bytes to store a date that can be stored in 2 bytes, at that point why not use 10 bytes?
String formatting and display seems more obviously affected to me than storage.