How do Greeks feel about having such a big and influential history?

Ambivalent. See:

Nick Nicholas’ answer to Do many modern Greeks feel a sense of failure or perhaps inferiority when compared with their ancient Greek ancestors?

The feeling has been there for a very long time. Theodore Metochites in the 14th century lamented that the Ancients had said everything that needed to be said, so there was nothing left for his contemporaries to do. The Greek peasantry would make up stories about the pagan giants who built the inexplicable structures all around them.

The more superficial have translated the feeling of inferiority into the bombastic (“When we were building Parthenons, you guys were eating acorns”—noone that feels secure in themselves bothers to say that to Westerners). The more sensitive have had the feeling of failure gnaw at them.

Dimitra Triantafyllidou’s answer to Do many modern Greeks feel a sense of failure or perhaps inferiority when compared with their ancient Greek ancestors?

It is a mixed blessing: a treasure and a burden. It all depends on the view of the person carrying the legacy. Being Greek can be seen as an enormous honor and an opportunity. People struggle to grasp things that are obvious at first sight. You set yourself the goal to reach high and embark on the journey like Odysseus. […]

There are Greeks who despair of the road and turn to Lotus-eaters. I am sorry for them. They have missed the journey.