Properly addressing this question involves dealing with a large amount of Jewish source material - about a quarter of the Mishna (by length) deals with the laws surrounding ritual impurity.
The short answer is no - we do not posses the "waters of separation" anymore, as you called it.
The rabbis decreed long ago that the land outside of Israel was considered to convey ritual impurity as a dead body since (among other reasons) it is possible that one may traverse an unmarked grave. It is debated in the Talmud whether this impurity is only through contact or even through proximity. Thus, anyone living outside of the land of Israel would be automatically considered impure as if they had touched a dead body.
This isn't particularly problematic, as this level of impurity needs cleansing predominantly to avoid the sin of entering the Holy Temple in an impure state. No temple, ergo no clear prohibition and no requirement to purify yourself before entry. There are other issues regarding tithing priestly gifts, but there are solutions to those that avoid infringing on biblical commandments.
The process of gathering the water requires a setup where the water carrier is guarded from birth from encountering death impurity (Parah 3:2), which is not something we do nowadays since 1) most people are assumed already impure and 2) as above, there's no specific need for the water since impurity itself is not a sin.
Technically, kohanim are forbidden from making themselves impure from having contact with a dead body and from participating in burial with non-family members. The prohibition here is the act the Kohen takes (or his lack of avoiding impurity), not the state of impurity itself.
The production of the ashes for the water appears to require the presence of the temple (3:8) which we no longer have. You promenade the heifer on ramp from the temple to Har Hamishcha (3:7) It also requires that the Kohen performing the ceremony be held in a special chamber in the Temple and sanctified with the waters of separation with ashes from a previous red heifer. (3:1)
Despite this last point, the law dictates that some of the ashes are preserved separately in order to perform the purification ritual under circumstances where the production of new ashes isn't available, such as in the current day. Unfortunately, I do not think anyone has found a cache of ancient red heifer ashes suitable for the reinstatement of complete ritual purity, nor is anyone setting up a "ritual clean room" to ensure there are children raised in complete purity to allow them to gather the water.
Even according to those opinions who state that we might be able to make the ashes without the Temple being built (that the portions of the Mishnah regarding the temple were historical and not legislative), no one has actually done so, and thus we do not have them in the modern day.