For medical insurance, look for a "major medical" policy only, which tend to be pretty reasonable and are usually available for up to 12 months +/-. They only cover catastrophic sort of things, like ER visits and the like.
Many renter's insurance policies can be continued for a period without having a physical residence. Usually they will pay a percentage of the overall coverage for things outside a residence. So, for example, if you have $100k worth of coverage, that might equate with $10-15k of coverage outside the residence. If you currently have a house or apartment, you might consider signing up for that insurance now before you move.
We used both these options in the past when we did long-term travel (by car). The renter's insurance also covered stuff we had in storage.