There is very detailed discussion about this issue in the field of Astrobiology. There is many constrains for "reasonable" life. I will try to go through most of them in short points.
Let me use following approximate definition of life in my answer:
Life is very complicated machinery of proteins (or other sufficiently complicated molecules or structures) that sustain themselves out of thermodynamical equilibrium in proper conditions, are capable of replication and of undergoing darwinian evolution.
To me, this seems quite general, and (I think) it does not trigger obvious counter-examples. For example fire is out of thermodynamical equilibrium, it is capable of replication, but does not undergo darwinian evolution nor consists of sufficiently complicated structures. Mule or other sterile animals are not alive per se according to the definition, but the environment in which they live and which produced them definitely contains life. This is a minor bug of this definition, but I do not know any better.
This criterion seems quite obvious. Fermions are elementary particles that follow the Pauli exclusion principle. Without it, everything collapses into one point or passes everything else and it is very difficult to imagine that there would be any reasonable complicated structured process involved. For example life based purely on electromagnetic waves is very implausible, since photons pass each other without noticing and they therefore cannot form structures.
Obvious candidates are electrons, which create atoms and molecules together with nuclei. But for example in the Forward's book Dragon's Egg, there are aliens inhabiting a neutron star and they are composed of neutrons and some protons in very dense degenerate neutron matter. This seems plausible in principle, since neutrons are fermions and could theoretically form structures in this environment.
Life needs liquids and membranes
If we investigate life based on chemistry, it seems life needs liquids. This is not an absolute criterion, however it seems that life as we know it strongly depends on liquid environments. There are incredibly complicated chemical processes in a cell, that require very precise conditions (pressure, concentrations of chemical compounds, density, etc.) In a gas, like in a nebula in space, density fluctuates heavily and it would be very difficult to do anything so complicated. Moreover, it seems that life needs to separate the relevant complex molecules it synthetized from the outer environment.
Plasma would be problematic too, since:
- It is as volatile as gas
- It is too hot for any complex molecules to be created
Metals and other solids have different problems. Complex molecules created in one part of the solid cannot be easily transported to another place, where they are required. In a cell full of liquid, every molecule floats around in a membrane or inside the cell and can randomly encounter other molecules and trigger chemical reaction. In solids, such mixing does not happen or would happen after very, very long time. (Of course now we are not talking about life looking like intelligent robots, that are assembled in factories. These would be probably plausible - but we are talking about life as a chemical process that could evolve on its own from more basic chemistry.)
Life needs complex polymers
If we investigate life based on chemistry, it seems it also needs stable polymers with rich chemistry. And here is a point at which carbon seemingly beats the alternatives:
- Carbon: Very rich biochemistry with many polymers with hydrogen, nitrogen, sulphur, phosphorus, and others. Bonds C-H, C-O, C-C, C-S, C-N have similar energy, which allows for big complexity.
- Silicon: Usually considered "the second best choice". Silicon, however, binds to oxygen much more strongly than to other atoms. This leads to a situation where either (on smaller temperatures) are silicon-oxygen bonds locked and cannot change, which makes things very difficult for complex chemistry, or (on higher temperatures) all bonds but the oxygen-silicon are broken and the complex chemistry must happen only between silicon and oxygen. (Silicones.) In absence of oxygen (which might be a problem, since oxygen is very abundant in the universe), and at very low temperatures, silane chemistry could be viable alternative for life. Zeolites at high pressures and temperatures seem like potentially viable option.
- Phosphorus, sulphur, boron, nitrogen: These elements form polymers, but they are very unstable. Life utilising them would have to work at very low temperautes, where it could be problem to find a viable solvent. (Life needs liquids). Moreover, these elements would much more likely form simple stable molecules than polymers, which would strip such life of a building material.
- Metal oxides: According to Wikipedia, oxides of metals have very rich chemistry, which could form a basis for life at very high temperatures and pressures (Like in Earth's mantle, for example).
As a conclusion: The universe seems to be quite restrictive about where life is plausible and what forms can it take. This list is not complete nor is it by any means certain, but it can give some idea what are the most probable candidates for life from the point of view of science.
Very good source can be found here - it is a short table listing polymer-forming elements and this is paired with typical solvent, that could take role of the liquid solvent in which the life based on given element could form. There is also another, even more extensive source on alternative biochemistries here.