In a public area you can not transport objects, the intention being that the Sabbath is a day of rest, and you should stay home or at least near your home and not do work.
Note: Jewish law very carefully defines "public area", and it would take several pages of text to fully explain it, so before objecting be aware you may be lacking background knowledge.
I’m guessing it’s the cubits that gave pause more than the restriction. I first became aware of eruvim thanks to an article in the L.A. Times which talked about a new eruv being constructed in Venice. The article talked about the restriction on carrying and how the eruv allowed congregants to bring their keys¹ with them while attending synagogue.
⸻
1. Doubtless someone reading this is thinking, well, they could just get an electronic lock and use that to protect their homes but most Orthodox rabbis would prohibit this as there is the possibility of a spark being created by the electronic system² which would violate the prohibition on making a fire on the Sabbath.
2. This question comes up in Surely You’re Joking Mr Feynman, where Richard Feynman, asked about this by rabbinical students, told them that electricity was not fire and that a condenser across the switch would eliminate any potential spark, but they were unsatisfied with his answer.
Keys on shabbat is doable without an eruv¹²³. (It's not considered carrying if it's an integral part of your clothing.) Eruvs are more for strollers/carrying small kids, dessert/wine when invited for lunch, etc.
Yeah, this is covered in the Eruv wikipedia article as well, but was not mentioned in the L.A. Times article (at least as far as I can remember—it’s been 15+ years since I read it).
Note: Jewish law very carefully defines "public area", and it would take several pages of text to fully explain it, so before objecting be aware you may be lacking background knowledge.