What was the status of specialized workers during Greek times?

Ancient Greece isn't as cohesive as Ancient Rome, each city-state had its own social structures. I'll concentrate this answer to Athens only, and try to give you at least another answer for Sparta.

The earliest known division of the Athenian society is ascribed to Theseus, the city's legendary founder, with three basic classes:

• Eupatridae

  The nobility, further divided in phatries, were probably the only full citizens at the time.

• Geomori

  The name suggests ownership of land, but not much is known about the class.

• Demiurge

  The lower class, craftsmen and labourers. The name literally means "makers" or "producer".

Draco, the first legislator of Athens, introduced the Council of Four Hundred and extended the franchise to all free men who could afford military equipment:

[Aristot. Const. Ath. 4.2] Citizenship had already been bestowed on those who provided themselves with arms; and these elected as the Nine Archons and the Treasurers, who were owners of an unencumbered estate worth not less than 10 minae, and the other minor offices from those who provided themselves with arms, and as Generals and Masters of the Horse persons proving their possession of unencumbered estate worth not less than 100 minae and sons legitimately born in wedlock over ten years of age. The new officials had to bail the outgoing Presidents and Generals and Masters of the Horse till the audit, accepting four sureties from the same rating as that to which the Generals and Masters of the Horse belonged.

[Aristot. Const. Ath. 4.3] And the Council was to be formed of four hundred and one members chosen by lot from the citizen body, and lots were to be cast both for this and for the other offices by the citizens over thirty years of age; and the same person was not to hold office twice until the whole number had been gone through, and then lots were to be cast among them again from the beginning. And if any Councillor, whenever there was a sitting of the Council or Assembly, failed to attend the meeting, he paid a fine of 3 drachmae if of Five-hundred-measure rank, 2 drachmae if a Knight, and 1 if a Teamster.

Solon, in the early 6th century BC, with his Constitution introduced a structured timocracy in Athens. Solon's timocracy is an early form of meritocracy and divided the population in four tiers, by measuring how many bushel of produce each man could produce in a year:

• Pentacosiomedimni

  The top class of citizens, whose property could produce 500 or more bushels of wet or dry goods per year.

• Hippeis or Triakosiomedimni

  The second tier of Athenian society, whose property could produce 300 bushels of wet or dry goods per year, roughly the cost of equipping themselves and one cavalry horse for war.

• Zeugitae or Diakosiomedimni

  The third tier of Athenian society, whose property could produce 200 bushels of wet or dry goods per year, roughly the cost of owning a hoplite armour.

• Thetes

  The poorest class, usually manual labourers who served in the fleet as oarsman, or in the army as light-armed support for the hoplites and the hippeis.

The Athenian Constitution was quite complex, here's a diagram based on Aristotle's observations:

The Pentakosiomedimnoi and the Hippeis were eligible for all positions of government:

• Archons
• Areios Pagos
• Boule (council of the 500)
• Heliaia
• Ecclesia (popular assembly)
• Officers in times of war (the Pentakosiomedimnoi could serve as generals)

The Zeugitae were not eligible to serve as Archons (or at the Areios Pagos, since its members were retired Archons), until 457/6 BC when archonship was opened to them. The Thetes only had access to the Heliaia and the Ecclesia.

Solon's timocracy is, however, only relevant to citizens (and more specifically adult males), while most specialized labourers at the time were Metics or slaves. Metics were mostly immigrants of Greek descent or freed slaves, held lower social status and were subject to taxation and enslavement for a variety of offences. They could be awarded isoteleia, a privilege that freed them of taxes and placed them above other Metics, and in very rare occasions full citizenship.

In Politics, Aristotle mentions that:

in some states manual laborers were not admitted to office in old times, before the development of extreme democracy

While mostly based on his observations of the Athenian society, there's also a relevant mention to Thebes:

At Thebes there was a law that no one who had not kept out of trade for the last ten years might be admitted to office.

Lastly, the hetaerae were highly educated courtesans, and perhaps the only women that were influential in Athenian society. While they often offered sexual services to their patrons, they held a high status, comparable to Geishas. Slave prostitutes of the era were known by a different name, p**nai, and were owned by a full citizen or a Metic.

The rest you can read in Aristotle's Athenian Constitution and Politics (especially the third book), and Theophrastus' The Characters, where through his outlines of thirty moral types you can deduce valuable information for the social classes of the era (for example, he lists pimps next to cooks as an ordinary profession). If you are interested in contemporary social commentary you should also look at Aristophanes' satirical plays, most (if not all) of which are deeply political and revealing of the Athenian society.