Ok, it seems I have been wrong about this as a general statement and liberal (or even most?) jew movements indeed consider it possible.
Last time I checked - it seems I read a viewpoint from a rather orthodox rabbi (but his article was the first one, that showed up in google at that time, now I cannot find it anymore), which clearly stated, this is not possible at all, with no exception.
The only way, would be to recognized as a "lost jew", meaning being of jewish origin, who lost connection to the tribe (even some generations ago). And the recognition would need years of devotion.
Maybe that is the case with some small, insular sects of Orthodox Jews? Some of the groups among those often referred to as the "Ultra Orthodox" - I don't know.
It is not aligned with the vast majority of Jewish views about conversion to Judaism.
In general, conversion is possible. As far as I can tell there are even clearly-enough defined requirements.
Jews are not supposed to treat converts any differently than non-converts. People being people, this doesn't always happen, but that is the reasonable principle.
Given that Jews for centuries have not proselytized to non-Jews, many believe you cannot convert to Judaism. You can.
I really cannot find that article anymore, but yes, it seems it must have come from an ultra ultra orthodox section and who knows why it ranked number 1 on google at that time (some years ago), fooling me. Because apparently yes, you absolutely can convert to judaism. It just isn't easy, like it is with other religions.
Last time I checked - it seems I read a viewpoint from a rather orthodox rabbi (but his article was the first one, that showed up in google at that time, now I cannot find it anymore), which clearly stated, this is not possible at all, with no exception.
The only way, would be to recognized as a "lost jew", meaning being of jewish origin, who lost connection to the tribe (even some generations ago). And the recognition would need years of devotion.