Are observations random variables?

Question

I'm looking at https://en.wikipedia.org/wiki/Central_limit_theorem and more specifically, at this part. When it says 'Let $X_1, X_2, \dots, X_n$ denote a random sample of $n$ independent observations', is it assumed that the $X_i$ are random variables? If so, I don't quite understand it because a random variable is a function from a set of possible outcomes to $\mathbb R$. But for example, if we're interested in measuring weights, $X_i$ is the weight of $i$th sample, which is a number and not a function. So the question is how can $X_i$ be regarded as a function? What is its domain (i.e., what is the set of outcomes in this case), and how is it defined?