Take a look at the documentation for Root. If a radical form exists ToRadicals can be used to perform the conversion.
sol = Minimize[{z, x/(1 + y) <= z, (1 - y)/x <= z, y - z <= z, 0 <= x <= 1, 0 <= y <= 1}, {x, y, z}]
sol // ToRadicals
If you want the numerical values
sol // N[#, 16] &
