I was wondering whether allenes can form rings; geometrically speaking, this seems like it would be energetically unfavorable due to their 180 degree linear geometry.
Yes they do. Such compounds are even found in nature, e.g. vernonallenolides.
As small as six-membered rings with allenes can exist, albeit fleetingly. Cyclohexa-1,2-diene has been known since a 1966 report by Georg Wittig (Angew. Chem. Int. Ed. 1966, 5 (9), 846), in which it is prepared by direct elimination of HBr from a vinyl bromide using t-BuOK:
For a more modern take on the matter, see e.g. Nat. Chem. 2018, 10 (9), 953–960 where Neil Garg and coworkers report the generation of azacyclic six-membered cycloallenes. The allene is generated by treatment of a 1,2-silyl triflate with CsF (this strategy will be familiar from benzyne chemistry). The cycloallene intermediates are trapped in various cycloaddition reactions.