This achievement is not hard but quite buggy. Many people, myself included, had trouble getting it, so I decided to make a guide for this achievement.. Here is some tips:

1) Play on beta version of the game. To do it right click on the game in your steam library, then choose "Properties", then choose "Betas", then choose "beta - Public Beta Test".

2) Disable steam cloud synchronization for this game (properties -> general -> bottom line) and delete file "Achievements.sav" in save folder (C:\Users\YourUserName\AppData\Local\RiverGame\Saved\SaveGames). You can make a backup if you want, just be sure you don't have this file in your save folder when you start a new endless game.

3) You don't need to delete other save files in save folder, so you can use items from dog's backpack from the previous game in your new game. Try to collect as much food and water as you can in dog's backpack before you die and start a new game. You can use any type of food (including meat). With 30 pieces of jerky and 15 jars of clean water you can spend at least 15 days in the starting location. You can use sleep option on campfire to save your time.

4) You need more than 20 days to get this achievement. I got it at the beginning of the 25th day. Some people said that they got it after death.
auri gave good advice in the comments

5) Try to avoid any type of animal death, even animal versus animal. It means don't use snakes or bears to kill other animals.

As I said earlier this achievement is buggy. No one knows what exactly you need to do to get this achievement. I just described conditions in my successful attempt (27.04.2018). Perhaps you don't need to follow all these steps, but better safe than sorry.

You can share your experience in the comments.