We've seen that there were

• ὁ ὄβολος 
• ἡ δραχμή worth 6 obols
• τὸ τάλαντον worth 6000 drachmas

But how did that money circulate, since there were no checks, plastic cards, banknotes?

In coins of course, well most of it. Silver coins at that, that's where the name for money comes from:

ὁ ἄργυρος, τοῦ ἀργύρου is silver, the metal
τὸ αργύριον makes it small (that's called a diminutive, it makes bigger things smaller, minute). Because money in Greece (at the time our story takes place) consisted of little silver pieces (coins), different size corresponding to different weight, which means different value.
Coins varied from a 1/4 obol (smallest) to a 4 drachma (biggest).
In paragraph 3 the ship's captain wants 5 drachmas, which for Dikaiopolis is far too expensive (he only has 6 drachmas in all, how could they live on 1 drachma during their journey?). They finally settle for 3 drachmas. So what coins changed hands?

As for τὸ τάλαντον, that only exists in accounts, there are no "talent" coins or notes in circulation, obviously, no widespread use of paper, so no notes.