Lower Steam's Pings per Minute. Go to Steam Settings, then In-Game, and find the "Server Browser Pings / Minute" dropdown. Change it from "Automatic (5000)" to "500." Some older routers cannot handle the high volume of pings the game sends out when searching for servers, causing the router to essentially freeze.
If you can play on the surface but cannot enter the caves, you are experiencing a or Firewall issue. DST uses two separate connections: one for the surface (Default) and one for the Caves Shard. don 39-t starve together online fix