a purely onchain DAO cannot own anything that isn't also own chain. In a multi interoperable chain world, a DAOs reach can be extended to other chains but still excluded from owning off-chain assets. in fact, a DAO doesn't know anything that doesn't happen on-chain. it needs oracles just to get read-only information about the outside world
there are two ways to get around this problem. the first is to use a legal wrapper, the LAO are pioneers in this effort. using a legal wrapper, a DAO can do anything a traditional entity can do plus all the onchain stuff a traditional entity cant. this is very cool and works great for a bunch of near term use cases. However, you are also burdened with the constraints of the legacy legal system and erodes most of the interesting properties DAOs bring
The second way which you eluded to is to use a decentralised arbitration service. Aragon has a protocol designed specifically for this use case.
quick TL;DR, Aragon is by far the best framework to deploy a DAO with today (disclosure, I work at Aragon :) before building the first MVP the founders realised there are a bunch of agreements that will never be computable in smart contracts. To solve this they planned to build a decentralised jurisprudence system that could take into account all the subtlety of human agreements and in a way that is crypto economically secure. you can find more information on Aragon, the court system and all the other amazing stuff we are doing in the links below.
If you're interested in DAOs, have an idea for a DAO or want to DAOify your existing organisation, you can find me in our discord server