Modern tokamaks can produce transport barriers TBs—localized regions with an increased energy confinement. Previous studies have been unable to examine the stability of internal TBs to radially extended short-wavelength magnetohydrodynamic instabilities “ballooning modes”, for the usual case with a sheared plasma flow and a magnetic shear that passes through zero near the TB. An established technique is adapted to study this situation, finding instability if 1 there is a low-pressure gradient, and if 2 the nearest “resonant surface” at which a Fourier mode is resonant, is sufficiently close. Surprisingly, flow shear is no more stabilizing than for magnetic shears of order one. This is explained. Without a strongly stabilizing mechanism, ballooning modes will fundamentally limit a TB’s radial extent, preventing them from extending across the entire plasma radius.