X-ray sources would turn to gamma rays. Not that it’s any better that X-rays. Other comments suggested lead plates. It would quickly get irradiated and would probably need to get shed as soon as you got to a destination.
No accounting for particles yet, which you'll also keep hitting, making your ship's materials radioactive and causing lots of secondary particle showers, bremsstrahlung and the likes.
First the particles will act like radiation, then they'll start causing matter-antimatter pair creation with your hull, then you'll get some exotic heavy quarks popping into existence, then you'll get some Higgs particles forming and at some point questions like "what is the mass of my ship" stop making sense.
No--x-rays will not make anything radioactive. Gamma rays can only do so when they're in the 2GeV range or higher (indirectly, through pair production.)
It's particles that make things radioactive, mostly neutrons as they aren't repelled by the nucleus and thus have a much easier time getting in. And while it would be hard to construct a shield out of it helium is effectively immune to becoming radioactive under neutron bombardment. And while lead isn't immune the reaction sequence produces nothing that won't be contained by the lead and it self-regenerates, enough bombardment returns it to where it started.
