Yes. It lives as a transaction output under op code 'OP_RETURN'.
See example of weather data stored on chain https://whatsonchain.com/tx/6dac238576e9778458bb62a2f7d936fd...
>are there fees to host this data (as we need to pay miners to include data in a block)?
Yes, current fees are about 1 sat/byte.
This is really a write once, read many type of system.