Claims

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1978_01_01_erdos: Erdős's statement, with the proof left to the reader, that a planar set of infinite measure contains a triangle of every prescribed area; later sources add unbounded sets of positive measure, so these sets answer the question.

2002_01_01_freiling_mauldin: Freiling and Mauldin's theorem, reported by Mauldin in 2002, that a planar set with no triangle of area greater than one has outer measure at most $4\pi/\sqrt{27}$; for convex sets it answers the question yes.

2013_01_01_freiling_mauldin: The result reported in Mauldin's 2013 survey that Erdős's constant $4\pi/\sqrt{27}$ is the sharp threshold when the set is the union of the interiors of at most three compact convex sets; the general case is open.

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