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Depends of the curvature of space time. Some scientists postulate that the universe is "closed", i.e. it is finite. So if you continued in a straight line in a particular direction, you would end up in the same place you started from, in a similar way that if you start in say New York and travel East, you will eventually circumnavigate the globe and end up back in New York again.
Others suggest that the Universe is "open". What that entails is a little harder to grasp, but in essence it means that, yes you could continue on in the same direction indefinately, and would never come to an "end".
To know for sure, we would nead some way to accurately measure the curvature of spacetime on the very large scale.
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