First of all, excuse potential misconceptions I have as I'm a layman. I'm aware that all of this is highly speculative.
According to eternal inflation, there are multiple bubble universes similar to ours separated by a de sitter space.
Did this entire multiverse originate from the inflaton? If yes where is this inflaton field even? Would that be an entirely new "space" and the multiverse is "contained" within?
First, a caveat, that all early Universe cosmology (pre-BBN) is speculative, don't trust anyone who speaks in absolutes about it.
Having said that, let's take the inflationary paradigm as given for this answer. Let's quickly review how ordinary, vanilla inflation is supposed to work (without eternal inflation and bubble Universes).
The usual story is that in the very early Universe, the energy density was dominated by a scalar field called the inflaton. Like any scalar field, the inflaton has potential energy, as well as kinetic energy (there is also "gradient energy", where the field can vary in space as well as time, but we will adopt the usual cosmological assumptions of homogeneity and isotropy, so gradient energy can be neglected).
Inflation -- or (approximately) exponential expansion of the Universe -- occurs when the inflaton's energy is dominated by potential energy. If the inflaton stayed exactly still on its potential, then it would effectively add a constant to the energy density of the Universe, and would act like a cosmological constant, which as we know from late Universe cosmology drives exponential expansion. However, the energy density during inflation, and therefore the expansion rate of the early Universe , was much higher than today.
The inflaton can't stay still on its potential, though, because the one thing we know for sure about inflation (if it did occur) is that it has to end, and produce a radiation-dominated Universe before big-bang nucleosynthesis starts. So the inflaton slowly rolls down a relatively flat potential. Eventually, it reaches a steep part of the potential, and the kinetic energy becomes larger than the potential energy, and at this point inflation ends.
You can think of the inflaton as a "clock" for the Universe in this period. The Universe expands and expands and expands, but there is a ticking clock until the inflaton reaches the part of its potential where it can no longer support inflation, and at that point inflation stops.
Eternal inflation is basically the same story, but we add another element: even though these are small enough to ignore gradient energy, there are small spatial fluctuations in the inflaton. So the "clock" will reach the "stop inflating" part of the potential at different times in different parts of the Universe.
Now you might think this is a small effect. The fluctuations are small, so the clocks will all arrive at "stop" at roughly the same time.
However, this situation is trickier than it first appears, because even if region $A$ has only a small amount more time to inflate than region $B$, because the expansion rate is exponential, region $A$ can end up much bigger than region $B$. So even though you might naively think it would be rare to find a region where the inflaton had many random fluctuations that happened to move the clock "away" from stopping (as opposed to the more common regions where there were fluctuations in both directions), that kind of region will end up having a very large spatial volume because it inflates for longer. This leads to the picture of "bubbles" (regions where the clock stopped) inside of a de Sitter Universe (an ever expanding region with fluctuations that keep bumping the clock away from stopping).
So, with all that in mind, here are my responses to the questions you actually asked:
Did this entire multiverse originate from the inflaton?
The inflaton is a component of the Universe. It drove inflation. Its also believed that the energy density of the inflaton decayed into standard model particles, through a process called reheating.
If yes where is this inflaton field even?
The inflaton field has long since decayed, and only is relevant at a much higher energy scale than we can probe experimentally.
Would that be an entirely new "space" and the multiverse is "contained" within?
In my answer I have been using the word "Universe" to describe "everything" -- both the observable Universe, and the parts too far away for us to observe.
What I would say is that, if eternal inflation is true, then a toy 2-d visual model of the Universe would be like a vast ocean with tiny islands in it. The ocean is constantly getting bigger and bigger. The islands represent the places where inflation has stopped. We would be on one of those islands. Our observable Universe would therefore be a tiny part of this vast ocean. However, in this analogy, the ocean and other islands are not in a different space than we are, they are in the same space but just further away. Similarly, in eternal inflation, the parts of the Universe that are still inflating and other "bubbles" where inflation has stopped are in the same spacetime as we are, just too far away for us to observe.
As an aside, the fact that the Universe is almost certainly bigger than we can observe should be uncontroversial. We don't expect the CMB to suddenly disappear in 100 years. What is controversial is basically, "how weird" the stuff can be that we don't observe. Should we assume it is largely similar to what we do observe, or is it possible the Universe would look very different if we could travel far enough fast enough to see it? That may or may not be a scientific question, but eternal inflation does give an answer to that question, and the answer it gives is that the Universe can look very different indeed outside of our horizon.
