According to everything we know today, including all experimental results and observations, nothing can be accelerated up to or beyond the speed of light. Light itself travels at that speed, and theoretically there's nothing against particles that are always faster (though we've never proven their existence). The problem is speeding something up to light speed and/or beyond. Relativity says that the faster something goes, the more massive it becomes (well known and measured, and even a necessary part of particle accelerator design). With more mass, you require more force to accelerate the object, and so ultimately at close to light speed, the object has nearly infinite mass and so requires infinite energy to be accelerated any more. Nearly every aspect of relativity has been experimentally proven so this result is pretty sound unless something new is discovered.

Any telescope, even the one Galileo used 400 years ago, efficiently collects the light from the universe, allowing us to see more clearly very distant and otherwise faint things. That light travels to our telescopes at the speed of light, and while very fast (300,000 km per second or 700 million miles per hour), it does take time to reach us from deep space. Sunlight takes a little over 8 minutes to reach Earth, the nearest star's light takes over 4 years to reach us, and from a distant galaxy it could take many millions of years to reach us. So we do see those distant galaxies as they appeared in the past, millions of years ago.