Short Answer: It's plausible, but we'll never know.
What follows is an analytical approach to this question. Different philosophies have different things to say about this sort of problem, so treat this as just one perspective.
This is a metaphysical question, and as such it's similar to many other unanswerable questions like the following:
- Is the world in my head?
- Are we all bumps on a log (or in The Matrix)?
- Is there a transcendent reality?
These questions make no testable predictions and so can be fit to any evidence. For instance...
- If I claim we're in a computer program and God comes
out of the sky and says we're not, I can just
claim that this too was part of the program.
- If I claim reality is in my head and you prove me wrong, I can claim your proof was also in my head.
And so on.
So first ask:
- Is there any evidence that can confirm or refute this?
If the answer is no, then the question is meaningless, and thus cannot be answered.
What if you're just asking if it's plausible that we are programs? Well it's plausible, but that's not saying much. After all, plausibility simply means that this proposition doesn't contradict existing evidence; the propositions above don't. It's easy for any proposition to fit the existing evidence if it says nothing testable.
Let's look more closely at some of the details of this claim. When we study the plausibility of this claim, we're asking if software can...
- Simulate reality.
- Host/achieve consciousness.
(1) is subject to processing power and storage, which is limited, but maybe the limits are enough -- especially if we consider things like parallel processing, quantum and grid computing.
(2) depends on the Hard Problem of consciousness, which is still controversial. Those who see consciousness as informational should have no problem with this.
But there's a deeper problem. How do we evaluate (1) and (2)? By looking at the state of our technology/knowledge. But we're not talking about OUR technology/knowledge, we're talking about those of whatever may be simulating us, which could be utterly different. In fact, limits on computing, consciousness and even physical laws could be arbitrary parameters of the simulation, as can be our cognitive limitations which can prevent us from seeing otherwise!
So what justifies these assumptions? Absolutely nothing.
And that's the problem. Without any kind of grounding, what can we say? We reason based on tautologies and/or observations, but once we postulate a super-sensible realm, there's nothing we can say about it, and no way to ground our reasoning. At that point, we can postulate anything, and there's no way to prove or disprove it and our assumptions are basically pulled out of thin air.
So yes, it's plausible, but this isn't saying much.
This is a fascinating question, and it can lead to interesting discussions and really broaden our perspective. However, that doesn't mean that it is a real question (one that can be answered now or in the future).
Metaphysical questions like these are a trap, and it's important to recognize them for the traps they are. In fact, Analytical Philosophy specializes in recognizing these kinds of traps and dispelling those kinds of questions, as it argues they can't be answered. Such philosophical "pseudo-problems" can't be resolved; they can only be dissolved. Or as Wittgenstein said:
Whereof one cannot speak, thereof one must be silent.
If you're interested in this kind of approach to philosophical problems, I recommend reading Language, Truth and Logic by AJ Ayer. He provides a very readable treatment of these traps, how they arise, and their implications.