Your question seems to talk about massive objects so I'll discuss massive objects. You have a misconception. If you start out $ 3 \cdot 10^9$ meters away from me, and head towards me at $.99c\approx 3\cdot 10^8$ meters per second (close enough for our purposes!), I will observe that you get here in 10 seconds. If I look at a clock you're holding (say you're a group of particles that have a specific half life, and when you get to me I'll measure how much of that original particle is left), I will observe that only $t/\gamma $ seconds have elapsed for you. In this case, $t=10\text{s}$ and $\gamma=(1-.99^2)^{-1/2}=10$, so only one second will have elapsed for you.
But this is absolutely symmetric. Nothing is accelerating, so you have full rights to look at the situation as if I'm the one traveling towards you. From your perspective, your time isn't dilated at all, it's my time that is dilated, and it's dilated by the exact same factor as before. So IF the distance between you and I from your frame was still $3\cdot 10^9$ meters, we would have a contradiction, because as discussed before my frame would observe that you had $1$ second of decay occur, while your frame would observe it took me ten seconds to get to you, so $10$ seconds of decay would occur. However, this isn't a contradiction because we haven't factored in length contraction!
From my perspective you're travelling at $3\cdot 10^8$ meters per second towards me, starting from a distance of $3\cdot 10^9$ meters. But from your perspective, I'm travelling towards you at a speed of $3\cdot 10^8$ meters per second, starting out at a distance of $3\cdot 10^9/\gamma=3\cdot 10^8$ meters. That speed at that velocity will get me to you in $1$ second. No contradiction.
All that this means is that you can get to the earth - from your perspective - as fast as you want. As you accelerate to relativistic speeds you'll find the distance between you and the earth shrinking (length contraction) and so the trip will seem to take almost no time at all, provided you accelerate enough. On earth, however, you get closer and closer to $c$ and may have more and more energy, but your speed stays approximately constant as you get very close to $c$ and so it will always take $d/c$ seconds, NEVER 5,000 years!
So, one might realize that to get really meaningful effects one shouldn't consider constant velocity problems. If you introduce acceleration, then you can get meaningful effects. Considering the twin paradox for example, one finds $t_{\text{earth}}=\gamma t_{\text{spaceship}}$. So the distance traveled (in Earth's frame of reference), if $v\approx c$, is $t_{\text{earth}}c=\gamma t_{\text{spaceship}} c$. If the distance to work with (to the moon and back) is a constant $d$, that is, if $\gamma t_{\text{spaceship}}c=d$, then considering only large very relativistic velocities with $v/c\approx 1$, $t_{\text{earth}}=\gamma t_{\text{spaceship}}=d/c$. The time will be constant again! But, of course, we already knew that, and didn't need relativity to answer it. To increase this value one must increase the distance. Increasing the velocity while keeping the distance constant won't help. While this only applies to massive objects, it might be alright to hand wave and say something like, "This is why the infinite time dilation of a photon doesn't matter when asking how fast does the photon get here". (in quotes because "infinite time dilation of a photon" is used here as a VERY hand-wavey concept.)
