How can I import Memberships without membership start date?
Let's assume that you can't edit the CiviCRM files or database  -  or only will do so as a last resort :)

These are existing memberships, so you should be able to export the memberships info with the actual start date, add the start_date to your CSV file and import with the start date. 

I'm surprised membership start date is required....I'll ask that question on stack exchange :)