When one visits historic sights in Greece, such as the Acropolis, are they approachable? What I mean is, can an individual (not in a large tour) actually enter some of the temples, such as the Apollo at Corinth, or the Parthenon, or are visitors kept at bay, allowed to only view from afar?
Regarding all 'major' temples, visitors are not allowed to enter them. According to the official Acropolis restoration service:
Why is it not possible to visit the interiors of the monuments?
There are two reasons why it is not permissible to enter the interiors of the monuments:
in order to protect the monuments from the additional weight that would be inflicted by the presence and circulation of visitors,
for reasons of safety, since interventions are being carried out on the monuments.
But you can get to see them from up close and are able to walk around them. Whoever is brave enough, can apply for a permission though (again, same source):
How may one obtain permission to photograph or make motion pictures in the interiors of the monuments?
Those interested may apply to the 1st Ephorate of Prehistoric and Classical Antiquities (A’ EPKA), Makryyianni 2-4, Athens and to the Acropolis Restoration Service. When the application has been reviewed, the applicant will be notified as to the relevant decision.
I would say there is no definite yes or no answer for all temples, because these are in various states. For a list of ancient temples, see here.
The Parthenon is undergoing extensive restoration work so you cannot enter it. You can get close enough to get a good view of the structure.