As far as California is concerned, your wages are taxable in California starting January 4th. Your living accommodations do not matter - it is the fact that you are working in California.
Yes, this means that while you may be considered a Michigan resident for all of January, you are a tax resident of California for California purposes.
Does this mean I have to pay taxes to MI and CA both for my stay since Jan 4th?
Yes, there may be some overlap. However, you can get a tax credit for taxes paid to another state on your resident return, so you will not be double-taxed.
Filing Tax Returns for Multiple States This link has instructions to handle your situation in TurboTax.