Yes, aleph numbers are so big that even considering them as a set leads to a contradiction. This is called the Burali-Forti paradox

https://en.wikipedia.org/wiki/Burali-Forti_paradox

It means that aleph numbers form a "proper class". That is a class, which is not a set. I'm pretty sure that implies them being bigger than any set, but that we also can't define their size.

There are many answers regarding this on stackexchange:

https://math.stackexchange.com/questions/1867467/is-the-set-of-aleph-numbers-countable

https://math.stackexchange.com/questions/1467/cardinality-of-all-cardinalities

https://math.stackexchange.com/questions/517173/are-there-countably-many-infinities