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This post gives

A stochastic process in discrete time n ∈ $N$ = {0, 1, 2, . . .} is a sequence of random variables (rvs) $X_0, X_1, X_2$, . . . denoted by $X = \{X_n : n ≥ 0\}$. ...

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what is the minimum ingredients to construct a stochastic process in discrete time?

  1. state space in discrete time, whose dimension $\geq 1$
  2. a sequence of random variables, whose length $\geq 1$

are these everything all you need to construct a stochastic process?

if yes, is a stochastic process the sequence $(X_0)$ which has only one elements? where the state space = {0,1}

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Besides the two ingredients you have mentioned, you also need a random variable. The random variable itself comes with a whole baggage of ingredients, which we are not listing for simplicity's sake. If the stochastic process is a sequence with only one point, then it generally takes the name "finite". A totally legal process though.

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  • $\begingroup$ In effect, you are just raising the same question again, because hiding behind the "random variable" is the need to define a meaningful sigma-field on the path space and construct measurable functions with respect to it. A consideration of this exposes the interesting issues. $\endgroup$ – whuber Aug 8 '19 at 11:14

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