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The first person to synthesize hydrofluoric acid in large quantities was Carl Wilhelm Scheele in 1771. This acid is known for its ability to corrode glass and metals.

What materials were the containers in which it was stored before the invention of plastics?

In the original 1771 experiment Scheele used a very simple setup consisting of a glass retort with the glass receiver (round-bottom flask).
Yes, the glass was etched to some degree by the fumes, but it was not that drastic to destroy the apparatus.

From Lennartson's The chemical works of Carl Wilhelm Scheele [1, p. 22]:

3.1 Publication 1. Investigation of Fluorite and Hydrofluoric Acid

Original publication: Kongl. Vetenskaps Academiens Handlingar. 1771, 32,
120–138.
Original title translated to English: Investigation of fluorspar and its acid

…

The second paragraph of the paper deals with the action of acids on fluorite, and
this is where the important discovery is made. Scheele mixed powdered fluorite
with sulphuric acid (oil of vitriol) and heated it in a retort (Fig. 3.3) whereupon corrosive fumes were emitted. He repeated the experiment with some water in the receiver. He found that a white crust was formed on the water surface, and that the water contained a dissolved acid. The inner surface of the retort was etched by the fumes. Scheele identified the white crust as silicon dioxide (siliceous earth) and the residue in the retort as calcium sulphate (selenite or gypsum). This lead Scheele to the conclusion that “Fluor [fluorite], therefore, consists principally of calcareous earth [calcium oxide], saturated with a peculiar acid; the nature of which we shall investigate hereafter”.

Fig. 3.3 Schematic view of the apparatus Scheele used for preparing “acid of fluorspar”

To address some critics and to proove the source of the white powder of silicon dioxide is the glassware it self, Scheele improved the setup, protecting the inner walls with tin and wax.
Further from [1, p. 98]:

3.60 Publication 60. Experiments with Hydrofluoric Acid

Original publication: Chemische Annalen für die Freunde der Naturlehre.
1786, part 1, 3–17.
Original title translated to English: New evidence for the characteristic nature of the acid of fluor spar

…

Scheele powdered fluorite in a brass mortar, mixed the powder with sulphuric
acid (oil of vitriol) and transferred it to a retort lined with tin. The receiver was lined with wax, and partly filled with water. This time Scheele obtained a distillate free from precipitated silicon dioxide (siliceous earth), and addition of potassium hydroxide gave no precipitate (i.e. no $ceK2SiF6$). Thus, this was pure hydrofluoric acid, HF (aq); similar results had been reported by Scheele’s friend, J.C.F. Meyer in Stettin.

References

1. Lennartson, A. The Chemical Works of Carl Wilhelm Scheele; Springer Berlin Heidelberg: New York, NY, 2017. ISBN 978-3-319-58180-4.