If there is no absolute time, how can we say the Big Bang was 13.8 billion years ago? The Big Bang is supposed to have happened 13.8 billion years ago.
Question: Given that there is no absolute time and different observers do not agree on the amount of time that has elapsed, how can we say that the Big Bang happened 13.8 billion years ago? Why wouldn't this kind of statement be ambiguous?
For example, say Alice stays on Earth for another billion years. She now measures the Big Bang at an (ostensible) 14.8 billion years ago. Meanwhile Bob goes zipping around the galaxy at 0.99c, and when he returns only 0.1 billion years of time has passed for him. His Big Bang is now 13.9 billion years ago. Both his and Alice's statements of the age of the universe are correct, making any statement about how many years ago the Big Bang happened ambiguous.
I feel like I'm forgetting something obvious here, but I can't place it. What am I missing?
 A: You are correct, there is no absolute time in the universe. This is as per SR, time is relative.
Now to whom is the universe 13.8 billion years old? To a comoving observer with zero comoving velocity (peculiar velocity). 

The "age of the Universe" of about 14Gyr you frequently hear about is a good approximation for any observer whose peculiar velocity is non-relativistic at all times. In practice these are the only observers we're interested in, since peculiar velocities for any bulk object (like galaxies) tend to be non-relativistic. If you happened to be interested in the time experienced by a relativistic particle since the beginning of the Universe, it wouldn't be terribly hard to calculate.

The age of the universe
Now this is only true for nonrelativistic observers. So you are correct, there is no universal time, but we usually use the CMB (Cosmic Microwave Background) as a background reference frame.
Understanding the CMB background as a reference frame 
A: Relativity has ways to make time pass arbitrarily slow, but no ways to make it pass arbitrarily fast. So if you are uncomfortable with defining what it means to "move with the universe", there is another way. You can define the age of the universe as the longest possible time any observer can need to get from the big bang to our current position in space-time. (Without cheating by using wormholes and other time-traveling shenanigans) The final result would of course effectively be the same.
A: First, you're quite wrong about Bob --- when he returns, he shares Alice's frame, and so agrees with Alice that the universe is 14.8 billion years old --- and that he once spent one billion of those years traveling, during which time he aged only .1 billion years, though this has nothing to do with how long ago the Big Bang occurred.
The person you want to ask about is not Bob, but Carla, who has been zipping along at .99c (relative to the earth) for several billion years, and is currently just passing  the earth.  She and Alice will certainly not share a time coordinate.
So it still makes sense to ask according to whom is  the universe 14.8 billion years old?  Roughly the answer is:  According to an observer who has been traveling with along with a galaxy.  In reasonably simple models of relativity, it doesn't matter which galaxy you pick.  The motion of the galaxies (plural) does give a global time coordinate (though there are still perfectly legitimate frames, like Carla's, that do not incorporate that time coordinate).  
Nobody believes those models are exactly accurate, but they can still be close enough to the truth that one can speak of the age of the universe as calculated in one of those models, and attribute some meaning to it.
A: In the right frame, all the galaxies are approximately stationary. That is - the relative velocity of the galaxies is mostly due to the expansion of space. So, the age of the universe can be computed consistently from the point of view of the galaxies in general, since they all share the same time frame. This is a contingent fact - if half the galaxies were at high speed with respect to the other, then you would not be able to define a single unique age of the universe. 
A: From https://en.wikipedia.org/wiki/Second#%22Atomic%22_second : "Since 1967, the second has been defined as exactly "the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom" (at a temperature of 0 K)." Assuming that the second is based of a physical measure and therefore has some reference to quantify it, I would say that time is not relative. Therefore, the second is just a small division of the calculated lifespan of the universe. Different time measurements will give you different time sizes. But the universe existence is absolute and it does not depend of a time measure to keep existing. If it had no inteligent life, the concept of time would not be present.
A: Concentrations of matter cause time dilation. The current hypothesis is that prior to the Big Bang all matter was "together" with no space between. This situation would mean that according to the laws of physics time was completely stopped. We could therefore say that time began with the Big Bang. Time itself has a value of 13.8 billion in terms of our years.
