In the case of ionic bonding, the answer would be that a stable structure is unlikely, although examples may exist among multimeric proteins. You may also check out the work of George Whitesides and others in the field of self-assembling soft matter.
However, to be clear, it is assumed that the arrangement in question is of alternating charges in a regular linear sequence (...+-+-+-+-+-+-...), that the attraction between nearest neighbors is isotropic, and that the arrangement is free of other forces (such as external fields). There could be a solvent but if it competes with the ionic interactions then stability is certainly impossible. Such an arrangement would be unstable in a vacuum as well except under exceptional circumstances. The reason is that electrostatic attraction operates over long distances, and charges will therefore experience attraction and repulsion from more than the nearest neighbors. In addition, since the bonding is assumed isotropic, there is little to interfere with bending under the influence of these forces and in the presence of thermal agitation. If one monomer can bond to more than two (coordination number>2) then a sheet or a nonlinear 3D structure will ensue.
However, if one monomer can bond stably to only two others, then an entangled, reptating polymer-like structure would result.
With regard to metallic bonding, there is such a thing as a "molecular wire". These can form electrically conductive single macromolecules. However I can't at present comment on whether bonding in these would be regarded as metallic, at least in a way comparable to bonding in traditional metal solids.
You might ask at matter modelling SE whether such models have been studied in silico.