They did farm. They did fish. They did hunt.

The main issue was that they did not know which of the English native crops were viable in Virginia growing conditions. The first year, they were instructed to grow test plots within the fort where they could be watched closely and kept away from scavenger animals.

In 1607, they had no expectations that the native Americans would be hostile. Bartholomew Gosnold had lead an expedition to the Cape Cod region in 1602 and had planned to stay there, but was worried about resupply, so they abandoned that attempt. The Native Americans there had been friendly and willing to trade. The Jamestown expedition had several people who had been on the earlier expedition.

Gosnold hedged his bets though. They knew they were not going to grow enough food, so he arranged for both trade and he had people in England planning and organizing steady resupply voyages. They slowly expanded their crops outside the fort, but it was not enough to feed the colonists and they needed to supplement with resupply.

The starving time was a direct result of the resupply missions. The third resupply mission was a fleet of ships carrying supplies and settlers. It ran into a storm and damaged the ships. Most of the ships survived, but one called the Sea Venture wrecked in Bermuda. They scavenged what they could from it. Another ship turned back. The remaining six ships continued on to Virginia. That added 200-300 settlers to Virginia roughly doubling or tripling the colonial population. But.. all of the food resupply had been on the ship that sank.

Basically, take a starving colony and add a lot more hungry people. That was the cause of the Starving Time.

Jamestown would had to be producing a surplus crop 3x the size of their colony to cover that.