Meaning
Ancient and outdated; also something incredible, absurd or extraordinary.
Example
Esse celular é do arco da velha, ainda tem antena!
That cellphone is ancient — it still has an antenna!
Origin
Arco-da-velha is an old popular name for the rainbow (arco-íris), and figuratively "coisas do arco-da-velha" means incredible, strange or prodigious things; from there it also colours anything that feels impossibly old. Dictionaries confirm the rainbow sense, but the origin of the velha ("old") element is disputed.
Sources: Ciberdúvidas, Priberam.
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