Can Greece exist purely as a service economy?
Theoretically yes; practically only with great difficulty. The more Greece becomes a service economy, the less products will be manufactured in Greece. However, Greeks will not consume manufactured products to any lesser degree than other people. Thus, the manufactured products which Greeks will want to have are going to be manufactured elsewhere and imported to Greece.
Obviously, if Greece’s services were offered abroad and generated income from abroad, things would be different. But to accomplish that, Greece would not only have to significantly increase exports (without manufactured products?) but also to approximately double the revenue from tourism (or generate other sources of foreign income).
Bottom-line: it is very unlikely that Greece will be able to manage without at least some focus on new domestic manufacturing.