The original Xbox 360 Arcade model (released 2007) cannot connect to Xbox Live because it lacks the necessary internal hardware—specifically, a built-in Ethernet port and the required memory for the network stack.
Later Arcade revisions (late 2008 onward) do support Xbox Live. Microsoft updated the hardware to include an Ethernet port and 256 MB internal memory (later 512 MB), enabling online functionality.
If you have an original Arcade without Ethernet, you cannot use Xbox Live. Upgrading to a later model (S or E) is the only solution.