Suppose I am traveling through space, accelerating closer and closer to the speed of light. As I speed up, photons traveling towards me become blue-shifted to higher and higher frequencies. When I am traveling fast enough, even what used to be visible light would all be shifted into very high energy x-rays or gamma rays. This seems like a big problem for my ship. Would my ship get torn/radiated pieces? How slow would I have to go to avoid this issue?
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Yes, but you'll have to go really, really fast. And even then, don't worry about the photons.
The relation between velocity $v$ and the observed and "true" wavelength $\lambda_\mathrm{obs}$ and $\lambda$ of the light is $$ \lambda_\mathrm{obs} = \sqrt{\frac{1-v/c}{1+v/c}} \lambda. $$ If you consider optical (i.e. visible to humans) light with a wavelength of, say, 5000 Å (green/bluish), then to be observed in your reference frame as a gamma-ray of, say 0.5 Å (roughly the wavelength of the softest gamma-rays) you'll have to go at $$ v = \frac{(1-\lambda_\mathrm{obs}/\lambda)^2}{(1+\lambda_\mathrm{obs}/\lambda)^2} c = 0.99999998 c. $$ The bulk of photons in space aren't optical, though. Optical photons are outnumbered by photons of the cosmic microwave background by roughly four orders of magnitudes (see this question on the number density of photons). In order to blueshift these photons to gamma-rays, you'll have to go at $v=0.9999999999999988c$.
At this velocity, your biggest worry won't be radiation: Let's start by assuming that you are in intergalactic space. Galaxies are dispersed in space with typically a few Mpc$^\dagger$ between them, or very roughly $L = 10$ million lightyears as measured by an observer in one of the galaxies. However, your extreme speed means that, in your reference frame, relativistic length contraction ensures that the average distance is $L\sqrt{1-v^2/c^2}\sim0.5$ lightyears, i.e. you will hit a galaxy once every six month. Galaxies are typically 100 kly ("kilolightyears") in size, or, in your reference frame, roughly two lightdays. Your risk of hitting a star is small, but you'll go through a column density of roughly $10^{20}$ hydrogen atoms per cm$^2$, all hitting you with almost the speed of light. I assume the cross section of your space ship is 10 m, so you'll be hit by $\sim10^{26}$ hydrogen atoms. Although this has a mass of only 200 g, its total energy is $$ E = \frac{10^{26}m_\mathrm{H}c^2}{\sqrt{1-v^2/c^2}} \sim 10^{23}\,\mathrm{J}, $$ or $\sim10^{18}\,\mathrm{J}\,\mathrm{s}^{-1}$, roughly corresponding to being hit by a pyramid at 20 km/s. Every second.
On second thought, you're probably not even going to make it to the nearest galaxy…
$^\dagger$1 Mpc = 1 million parsec ~ 3.3 million lightyears.
