Do I need to earn reputation of 200 points per day consecutively for 50 days and 150 days to earn the Epic and Legendary badges?
For the Epic badge, you need to hit the daily reputation cap on (at least) 50 days. I.e. you need to have 50 days in your SO history when you received at least 200 rep points. The days don't need to be consecutive though.
For the Legendary badge, the same rules apply, except that the limit is 150 instead of 50.
Note that you can check at the bottom of your rep audit page, how many days you have reached the daily limit.
No, it's not consecutive. I recently received the Epic badge on SO, and I can assure you I didn't break the limit on 50 consecutive days.