The stress-energy tensor has units of energy density. Indeed as we usually write it the $\mathrm{T}_{00}$ elements is just the energy density i.e. joules per cubic metre.
You are quite correct that the other three diagonal elements $\mathrm{T}_{11}$, $\mathrm{T}_{22}$ and $\mathrm{T}_{33}$ behave like a pressure. For more on this see Intuitive understanding of the elements in the stress-energy tensor. However pressure and energy density have the same dimensions. Pressure is force/area. Multiply the top and bottom by distance and you have force $\times$ distance/volume, and force $\times$ distance is work i.e. energy.
The other elements are a momentum flux i.e. a momentum density times a velocity. Again if you work through the dimensions you'll find this has the same dimensions as an energy density.