I had this thought in my head for a couple of months now and I really wanted to see what the answer to it is. So here it is:
A man is sitting in a white room, where all he has to do is cross over a purple line to travel back to Earth. However, he has infinite time to do so. Does this mean he has a 100% chance of crossing the line, because he has as much time as possible, meaning that soon enough sometime he will cross the line?
(Also not accounting for eating, bathroom, or sleeping)
This can be answered somewhat precisely using the theory of random walks. We could model the man's motion as a random walk in the 2D-lattice $\mathbb{Z}^2$, starting at the origin $(0,0)$. It turns out that in one or two dimensions, the probability of eventually passing through an arbitrary point $(a,b)$ tends to $1$ as time goes to infinity. Interestingly enough, this doesn't hold in three dimensions, where the probability of reaching arbitrary $(a,b,c)$ is only $\sim 34 \%$ on average, and this decreases the more dimensions you add.
So your hypothesis is correct - given enough time, the man would eventually cross the line. Of course, this assumes the man in question is constantly moving randomly, which is a questionable assumption in itself...