The most important purity-related reason why they can't perform sacrifices is that they can't enter the temple mount due to impurity-related laws, but ways to cleanse that ritual impurity do exist - they're just not readily available (and haven't been since about 100CE). The temple having been sacked by the romans is also kind of an important impediment. (And as they can't enter the temple mount, they can't rebuild it... and there'd be quite some political implications to such a rebuilding...)
EDIT: furthermore, most of modern Jewry (more than 98%) derive from rabbinic Judaism - this includes orthodox, conservative, reform and reconstructionist Judaism. The forms that still think the temple is important- viz. orthodox (and conservative?), think that they need to know the exact location of the altar within the temple premises, etc. The (ultra-)orthodox expect the prophet Elijah (who they expect to be returned from the heavens), and the Messiah, to help out in pointing out where it should be. Until then there's no point at all in even trying.)
Peter Brown wrote:I think I see. There are no more people left that were picked by god to perform the offerings, so they can no longer be done.
The cohanites and levites still exist, certainly. There's even a fair share of genetical research that shows that most who self-identify as cohanites do share a common ancestor. It's just that they don't have a temple at the place the sacrifices are supposed to be conducted, nor are they currently ritually pure so they can't enter it.
I used the word rabbi just as some might use priest. But I think youre saying there are no Jews able to be priests anymore.
The rabbi, basically, is more like a scholar (and judge in a kind of legal system, that of halaka) than he is like a priest (of the catholic or orthodox christian kind) - he has no special ritual function. (Though it is not unknown that a rabbi may be the only one available to have the requisite knowledge for some rituals, altho' e.g. the liturgy is generally performed by a chazzan, a cantor, but can be lead by any Jewish boy that knows the liturgy and is over 13, in conservative synagoges women can also lead a mixed congregation in prayer). The Levites and Aaronites still exist, but their ritual function is very very circumscribed. (Also, it's not necessarily the case that everyone that is from those subtribes really have the status required, some actions can cancel one's eligibility so that even one's descendants won't have it)
The Samaritans, a schism out of Judaism that occured like half a millennium bce, still perform some sacrifices - but they consider Gerissim to be the place where God wants sacrifices, and they, afaict, consider the need to perform sacrifices greater than the need to be ritually pure when performing them (or they have different rules for how to become ritually pure, but as that is described in the actual pentateuch, I'd think they wouldn't differ that much).