Our source for this story is Vitruvius, and I have yet to encounter a single person who has actually bothered to read the passage. If anybody actually did bother to read the passage some items that are usually taught (are even in standard works on the history of mathematics!) would be understood as being without a source. For example everybody seems to think the gold crown Archimedes was tasked with assessing was Hieron's own, which in fact Vitruvius says was to be a temple offering. The Latin reads:

Tunc is, cum haberet eius rei curam, casu venit in balineum, ibique cum in solium descenderet, animadvertit, quantum corporis sui in eo insideret, tantum aquae extra solium effluere.

Then he, when he had been tasked with the matter, came by chance into the bath, and there when he lowered himself into the tub he noticed that as he settled himself into it so the water flowed out of the tub.

There are several things here, and none of this is really clear enough to say whether this is a private or public bath. The Greeks did have public baths, especially in the great centers of the Hellenistic Period, when such structures became relatively common. Greek baths were arranged somewhat differently than their Roman counterparts--instead of a few large communal pools Greek baths generally featured a large number of small hip baths, like those at the public bath at Gortys in Crete. Most Greek public baths did not have complex heating systems, although the one at Syracuse (and the one at Gortys) did have extensive underground heating chambers. Balneum is usually plural when referring to public baths, although not exclusively and it's much less unusual to see it in the singular if Vitruvius is emphasizing the actual vessel or its Greekness--in Greek these structures are singular, βαλανεῖον. Likewise Vitruvius says casu venit, "he came by chance," which seems a somewhat bizarre statement if we're talking about Archimedes' own home, and Latin regularly uses verbs of motion along with in balnum or in balneas to indicate the action of heading to the public baths. The real smoking gun is that Vitruvius says that Archimedes then ran back home: nudus vadens domum. So much for this being a private bath.

Even so, even if this were a private bath there's no problem here. The large, spacious baths of the Roman wealthy required large quantities of water (easily supplied in the Roman world by private connections to the aqueducts) and continuous fuel to keep it warm. The baths of the Greek world, and those of the less affluent at Rome, were totally different. Full tub baths had largely disappeared in Greece by the Archaic Period, and the Greek world had never known anything like the enormous basins of the Romans, more like swimming pools than tubs. By the Classical Period and throughout the Hellenistic Period most baths in the Greek world were hip baths, which required relatively little water and could be filled and refilled easily without any complex heating apparatus. This is certainly what Vitruvius is describing. Only in such a basin could Archimedes have noticed any overflow resulting from his entry, unless we suppose a much larger swimming pool sized vessel overfilled--even then, any trip to the pool reveals that any body of water that size (especially with other people in it) is bound to have ripples and other motion that would affect observation. The Latin used of the tub is solium, which means a (usually backed) chair or throne. It can be used of bathtubs as well, but generally these low hip baths, not a basin of any great size. Vitruvius' language is a little odd as well if he's imagining anything larger than a hip bath. It's a little bit odd (though not insane) to see insideret here--insideo can mean "to settle down into," but it really means "to sit on," from sedeo, "to sit." Not really appropriate (or usual) for entering a tub of any great size, but precisely the correct word for describing a hip bath.

Also, why wouldn't Archimedes be wealthy enough to afford a bath, even if the Greeks had used full-sized basins? Archimedes was the son of a court astronomer and was the court intellectual of the Syracusan state under Hieron and Gelon. He received extensive patronage from them and had correspondence with the royal Alexandrian scholars--all this at a time when being an intellectual at a Hellenistic court was among the most secure jobs in the world. Why should we presume he would have had any significant monetary issues?