I think there are few different questions you might be interested in, that are worth separating out.
1. "Real numbers"/"decimal expansions"
The phrase "Real number" means a number of very specific (equivalent) things, and the same is true of the phrase "decimal expansion". Under these strict definitions, the answers to all variations of your question are "no, we can't do that".
For example, whether we define the real numbers as decimal expansions (as alluded to by Tim Gowers) or we define real numbers in some other way and define decimal expansions on top of them: we would intentionally define things so that the sequence of digits after the decimal point is the usual sort of sequence where there's a digit for each natural number, and we can't speak of things like "a digit after a digit for each natural number" as described in the OP.
For another reason why we can't do something like "smallest positive real number", see #3 below.
2. Some other expansion?
Well, what if we throw away the rigidness of "real numbers" or "decimal expansions". Can we form something along the lines of "0.000...(Aleph Null zeroes)...0001"? The answer is "yes, in at least two different ways".
One clarification that probably applies to all ways you can do something like this: Note that order matters here; "$0.1\underbrace{00\cdots}_{\aleph_0\text{ zeros}}$" should probably just be $0.1$, but "$0.\underbrace{00\cdots}_{\aleph_0\text{ zeros}}1$" should not. Because of that, it's unconventional and potentially misleading to write the cardinal $\aleph_0$. We really want the ordinal $\omega$.
One literal approach discussed in a few places is to sort of declare by fiat that we can use other ordinals to index digits. The result is referred to in a few places as "Kaufman decimals". This is explored loosely/recreationally here and a bit more rigorously here and elsewhere. I'm not too familiar with this approach, but my understanding is that while you can probably get a fairly reasonable ordering of positive numbers in this fashion, you probably run into trouble making nice definitions for negative numbers or multiplication.
Another thing that is closely related to the idea written in the OP is (Gonshor's) sign expansion for the Surreal Numbers. It almost matches binary expansions in the simple cases, but extends very well into higher ordinals. I describe it briefly in this answer to Construction of an infinite number type and other ideas.
3. Smallest positive in some other number system?
Ignoring the decimal expansion stuff, can we have a smallest positive number in some number system other than the reals? It depends on what properties you want. I basically answered this question in another question which was closed, so I'll re-present things here.
If you just demand something like ordering that plays well with multiplication and subtraction, then the integers should be okay. And there is certainly a smallest positive integer: "$1$".
But if you have a number system where you can divide as well (an ordered field), then there can't be a smallest positive number for the reason in Chris Culter's comment. If $x>0$, and the order plays nicely with the operations, then we have $\displaystyle{x>\frac{x}{1+1}>\frac{0}{1+1}=0}$. In other words, dividing a positive by $2$ always gives a smaller positive.
The question remains: is there any previously-defined or otherwise-justified context where a fraction like $\frac12$ would make sense, but where there is still a smallest positive thing (you may not want to call it a "number" for the above reason)? There is, but it's really esoteric.
In combinatorial game theory, when studying games that might go on forever, there are games that have "values" which correspond directly conventional numerical values like $\frac12$, as well as a game called $\mathbf{tiny}$ whose "value" is the smallest positive one across all games. In this context, you can't multiply every pair of games, or even compare every pair of games in the ordering, but we do have this smallest positive game. For a little detail about how that is (though I admit it doesn't explain things terribly well), see this answer of mine to Is there a symbol for the idea of the smallest value greater than zero?.