Not a barrel. Diogenes did not live in a barrel. The barrel that flits around English literature and high school classes is a mistranslation of πίθος, a type of very large pot. It's mentioned at Diogenes Laertius 6.22:

ἐπιστείλας δέ τινι οἰκίδιον αὐτῷ προνοήσασθαι, βραδύνοντος, τὸν ἐν τῷ Μητρῴῳ πίθον ἔσχεν οἰκίαν, ὡς καὶ αὐτὸς ἐν ταῖς ἐπιστολαῖς διασαφεῖ

And he had ordered someone to get a house for him, and when this man delayed he dwelt in a wine-jar in the Metroön [in Athens], as he explains in his letters

Diogenes didn't just live in a wine-jar for the sake of living in a wine-jar, he did so, at least according to Diogenes Laertius' apocryphal story, because he had nowhere else to live. A great deal of what we "know" about Diogenes is pretty much found only in Diogenes Laertius, who's not a particularly good source. This is no exception--the New Pauly points out the difficulty in describing his life with the note that "truth and legend are mixed throughout his biography." At some point he lived with Xeniades in Corinth, who purchased him after he was sold into slavery--Diogenes Laertius quotes Eubulus as claiming that Diogenes died in Xeniades' house an old man. At any rate, Diogenes' wine-jar was just an ordinary wine-jar, and Diogenes Laertius mentions that it was broken:

μειρακίου γοῦν τὸν πίθον αὐτοῦ συντρίψαντος, τῷ μὲν πληγὰς ἔδοσαν, ἐκείνῳ δὲ ἄλλον παρέσχον

Anyway, when a young man smashed his wine-jar they [the Athenians] gave him a beating, but gave Diogenes another wine-jar.

On the off-chance that Diogenes' wine-jar (provided it existed), an essentially disposable item, had survived, how would we know it? What are the chances it would even have survived intact? And moreover, why would anybody be looking for it?