I have also seen Star Trek as a boy, where there is a silicon-based lifeform from Janus VI (the Horta), which eats through rock like we walk through air. There's also a great Marvel character, the Sandman, who can transform into glass and back, and can manipulate the Earth - a marvellous ability. But, the short answer is that silicon-based life is probably restricted to the realms of Sci-Fi.
The idea is broadly based on silicon's apparent similarity to carbon. Silicon is close to carbon in the periodic table, it has a valency of 4 (the same as carbon) and so can form bonds with 4 other atoms and could therefore apparently be a building block a bit like carbon - and so may in principle form similar structures, one might think. It is also widely available in the biosphere (indeed, in the universe) as a potential building block, in quantity.
However it all falls apart from there.
"For life to survive the agent(s) must be universally and continuously available, in quantity, and be a replenishing not a diminishing resource"
Life is driven by some from of an energy source or sources, and needs an agent(s) and process(es) to release and use the energy available in its energy source(s). For life to survive the agent(s) must be universally and continuously available, in quantity, and be a replenishing not a diminishing resource. In our environment we have oxygen available to be that agent, and during cellular respiration glucose (as a carbon source) is broken down to produce CO2 and H2O, and energy is released. One could go on about ATP, and O2/CO2/various cycles and so on, but that's not the main point here, really.
The main point is that for life as we know it, when carbon reacts with oxygen it can form gasses - for example CO2 - and water, both of which are easily eliminated from a carbon-based organism in the process of its metabolism.
"A silicon-based life form would not be able to get rid of the waste products arising from its own metabolism. It would therefore be a severely self-limiting life form"
In comparison, when silicon reacts with oxygen, it forms a stable lattice which is then not possible to easily eliminate from the "organism" (if the organism did indeed exist), and so a silicon-based life form would not be able to get rid of the waste products arising from its own metabolism. It would therefore be a severely self-limiting life form!
And, that's ignoring the implausibility of the necessary oxygen replenishment processes and the other various cycles required for sustaining silicon-based life if there is no CO2 and water release, and from which we benefit with carbon-based life.
The theoretical biochemistry of silicon is also problematic since it does not form many chiral compounds (despite considerable efforts in the laboratory); with carbon compounds, energy is released in a series of tightly controlled, interlocking steps whereby "right handed" carbohydrates are oxidised by enzymes composed of "left handed" amino acids. (Interestingly, it has been hypothesised that the handed-ness is a result of the the first carbohydrates forming on a silicon surface, thus the handed carbon system that evolved was perhaps initially dictated by silicon, which one would think might be a bit perverse in all the circumstances. But that's life!)
However, the overall summary of all this is that energy-release from a hypothetical silicon-based life form is a massively problematical theoretical issue, which one would think would preclude the life form itself existing.