We do not really know what the interstellar medium looks like up close and personal. If it is just stray hydrogen atoms you will just experience a head-on flow of 'cosmic rays' that will collide with your spacecraft and probably generate secondary radiation in the skin of your ship. This can be annoying, but it can be shielded so long as the particles are not ultra-relativistic. At speeds of 50-90% the speed of light, these particles are not likely to be a real problem. At speeds just below the speed of light, the particles are ultra-relativistic and would generate a very large x-ray and gamma-ray background in the skin of your ship.
As it turns out, our solar system is inside a region called the Local Bubble where the density of hydrogen atoms is about 100 times lower that in the general interstellar medium. This Bubble, produced by an ancient supernova, extends about 300 light years from the Sun but has an irregular shape. There are thousands of stars within this region which is enough to keep us very busy exploring safely.
Interstellar space also contains a few microscopic dust grains (micron-sized is common) in a region about a few meters on a side. At their expected densities you are probably in for a rough ride, but it really depends on your speed. The space shuttle, encountering flecks of paint traveling at 28,000 mph (about 6 miles/second or 0.005 percent the speed of light) is pitted and pierced by these fast moving particles, but dust grains have masses a thousand times smaller than the smallest paint fleck, so at 0.005 percent light speed, they will not be a problem.
At 50 percent the speed of light which is the minimum for interstellar travel you will cover enough distance in a short amount of time, that your liklihood of encountering a large interstellar dust grain becomes significant. Only one such impact would be enough to cause severe spacecraft damage given the kinetic energy involved.
A large dust grain might have a mass of a few milligrams. Traveling at 50% the speed of light, its kinetic energy is given non-relativistically by 1/2 mv^2 so E = .5 (0.001 grams) x (0.5 x 3 x 10^10 cm/sec) = 1.1 x 10^17 ergs. This, equals the kinetic energy of a 10 gram bullet traveling at a speed of 1500 kilometers per second, or the energy of a 100 pound person traveling at 13 miles per second! The point is that at these speeds, even a dust grain would explode like a pinpoint bomb, forming an intense fireball that would melt through the skin like a hot poker melts a block of cheese.
The dust grains at interstellar speeds become lethal interstellar 'BB shots' pummeling your spacecraft like rain. They puncture your ship, exploding in a brief fireball at the instant of contact.
Your likelihood of encountering a deadly dust grain is simply dependent on the volume of space your spacecraft sweeps out. The speed at which you do this only determines how often you will encounter the dust grain in your journey. At 10,000 times the space shuttle's speed, the collision vaporizes the particles and a fair depth of the spacecraft bulkhead along the path of travel.
But the situation could well be worse than this if the interstellar medium contains lots of ice globules from ancient comets and other things we cannot begin to detect in interstellar space. These impacts even at 0.1c would be fatal...we just don't know what the 'size spectrum' of matter is between interstellar 'micron-sized' dust grains, and small stars, in interstellar space.
My gut feeling is that interstellar space is rather filthy, and this would make interstellar, relativistic travel, not only technically difficult but impossible to boot! Safe speeds for current technology would be only slightly higher than space shuttle speeds especially if interstellar space contains chunks of comet ice.
This is an issue that no one in the science fiction world has even bothered to explore! The only possible exception is in Star Trek where the Enterprise is equipped with a forward-directed 'Brussard Deflector' (that big gold dish just below the main saucer) which is supposed to sweep away particles before they arrive at the ship. This is very dubious technology because hydrogen atoms are not the main problems a ship like that would have to worry about, especially traveling inside a planetary system at sub-light speeds.
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