By making the neder to bring a mincha with barley when he knows it is not possible, this person is making a neder in vain – a neder shov.
From INSIGHTS INTO THE DAILY DAF Nedarim 15 I see that
The RAN (end of Nedarim 14b) writes that there is no Isur of "Neder
Shav," a Neder made in vain, as there is an Isur of Shevu'as Shav, a
Shevu'ah made in vain.... (it means that) there is no Neder
So he does not have to bring anything.
TOSFOS in Gitin (35a, as cited by the GILYON HA'SHAS) and
the ROSH here rule that although Malkus is not administered for making
a Neder Shav, one still is not allowed to make a Neder which he cannot
fulfill. The verse says, "Lo Yachel Devaro" -- "he shall not profane
his word" (Bamidbar 30:3). One who makes a Neder that he cannot
fulfill transgresses the prohibition against profaning his word.
(He is not punished with Malkus because his transgression is a "Lav
she'Ein Bo Ma'aseh" (it involves no physical action). Although a
Shevu'as Shav is also a "Lav she'Ein Bo Ma'aseh" and yet one is
punished with Malkus for transgressing that Lav, this is because a
special verse in the Torah teaches that Malkus is given for the "Lav
she'Ein Bo Ma'aseh" of Shevu'as Shav.)
At all events he does not need to bring a mincha.