To access any data outside the Ethereum blockchain you need a way to get the information into the blockchain. And the only way to do that is by transactions. So either you have to send the transactions yourself or organize for an oracle to send them for you.
Without really knowing how EOS operates, the opposite is probably also true for EOS. It's a deterministic blockchain so all data has to come through transactions. EOS probably has its own oracle-like services which provide the data.
If oracles support it, you can organize for them to read data directly from the other blockchain. If oracles don't support reading directly from another blockchain, you need to use some other extra services between. Then the oracle just writes the data into the blockchain and provides it for your smart contracts to access.