Do water barrels for passive solar heat storage actually work in a greenhouse?

They do, within limits worth being clear about. Water has an unusually high specific heat capacity — around 4.2 kilojoules per kilogram per degree — so a given volume stores far more heat than the same volume of gravel, brick or soil. A hundred litres of water cooling by five degrees releases roughly two megajoules overnight, which is meaningful in a small structure.

To get that, the barrels need sun on them. Place them along the north wall of an east-west oriented greenhouse where they intercept low winter sun through the south glazing, and paint them matt black so they absorb rather than reflect. Dark containers with a large surface area relative to volume charge and discharge faster; a few large drums store more total heat but respond more slowly than several smaller containers.

What they realistically achieve is smoothing rather than heating. Expect them to raise overnight minimums by one to three degrees and to reduce the daytime peak, which is itself useful — wide temperature swings stress plants more than steady cool conditions do. They will not keep a greenhouse frost-free through a still, clear, minus-eight night, and marketing that suggests otherwise is overselling it.

Gravel or stone beds work on the same principle but need roughly three times the volume for the same storage, which is why water is generally the better choice where you have the space for barrels.

Two practical notes: add a small amount of algaecide or keep the containers sealed and opaque, or you will be growing green sludge by March; and site them where a leak would not matter, because a split drum indoors in February is an unpleasant surprise.