How did the south manage to fight off the north for 4 years during the civil war?

Because they had certain advantages over the Union:

Some of the best officers of the US Army had resigned to join the Confederates.

Traditional advantage in cavalry - most young Southerners could ride and shoot

Officer class - the army was a more natural choice of career for sons of rural landowners than in the industrial North.

Rank-&-file - a more soldierlike breed, a generally aggressive kind of people

Home-ground advantage - most battles were on terrain unfamiliar to the enemy

Stronger war-mission - to defend the homeland against the invader.

Their President did not have to face an election in mid-war, as Lincoln did.

Sherman was one of the few who guessed it would be a long war. But he said that Union victory would be inevitable in the end.