Although buckling is a prime route to achieve functionalization and synthesis of single colloids, buckling of colloidal structures-made up of multiple colloids-remains poorly studied. Here, we investigate the buckling of the simplest form of a colloidal structure, a colloidal chain that is selfassembled through critical Casimir forces. We demonstrate that the mechanical instability of such a chain is strikingly reminiscent of that of classical Euler buckling but with thermal fluctuations and plastic effects playing a significant role. Namely, we find that fluctuations tend to diverge close to the onset of buckling and that plasticity controls the buckling dynamics at large deformations. Our work provides insight into the effect of geometrical, thermal and plastic interactions on the nonlinear mechanics of self-assembled structures, of relevance for the rheology of complex and living matter and the rational design of colloidal architectures.