No. It can’t and you couldn’t.
You can accelerate forever and never reach the speed of light because of relativity. I know a lot of people here are answering differently, but this is the correct answer.
If you had unlimited fuel you could theoretically accelerate at 1G forever. Therefore you walk around your spaceship with normal “gravity.” However, you would never achieve the speed of light and you would never exceed the speed of light.
When you got back to Earth, however, people might seem old.
Let me see if I can’t put this in simple terms. You are the center of the universe for you. Exactly as they thought in the olden days. Shine a light in any direction and the light goes away at the speed of light for you. “Speed of light for you” implies a speed, which is relevant to the observer.
You are in a spaceship. Never mind when it left. Never mind how far the burners burned. Never mind if your are still accelerating. You are in a spaceship. You go to the bridge and look around. What do you see? Stars.
It’s not like Star Trek.
You don’t see the stars moving by.
You see stationary stars. To get to any one of them is going to take you a long, long time.
From your pocket you pull out a flashlight. You point it in every direction and turn on the beam. The beam leaves your flashlight at the speed of light. For you. That’s the sum and substance of Einsteins theory of relativity.
Back to your question. If you keep on accelerating, can you exceed the speed of light. No! Of course not. You are in space, looking around. Who knows how “fast” you’re going. You are in space surrounded by points of light you know to be stars. However you got there is not important. To go this way or that way you are going to have to burn fuel which will cause you to accelerate.
Space is relative. Time is relative. Spacetime is relative.