If I flip a coin 1000 times in a row and it lands on heads all 1000 times, what is the probability that it's an unfair coin? Consider a two-sided coin. If I flip it $1000$ times and it lands heads up for each flip, what is the probability that the coin is unfair, and how do we quantify that if it is unfair? 
Furthermore, would it still be considered unfair for $50$ straight heads? $20$? $7$?

You asked 

if this specific string of flips happens, what is the probability the coin is unfair?

And it seems to me that most answer here are answering the question 

if the coin was fair, what is the probability of this specific string of flips happening? 

If I try to literally answer your question, I get stuck unless we make additional assumptions. For your question, the sample space would have to be something like all instances ever of flipping a coin 1000 times. Not one specific coin mind you, but all instances ever, anywhere, of flipping one coin 1000 times. The next instance of 1000 flips can use a different coin. Some elements of that sample space would have 1000 heads. And some elements of that sample space would have involved a fair coin.
But we can't do any calculating until we understand, among coins, what is the distribution of fairness? $p$ for a coin can range form $0$ to $1$, with only exactly $0.5$ being fair. If you assumed a specific distribution for the fairness of all coins, then we'd have the necessary information to calculate an answer for your literal question. 
In a world where every coin is fair, then even with these 1000 heads, there is a 0% chance that your coin is unfair. In a world (more like the real world) where no coin is truly fair, then there is automatically a 100% chance that your coin is unfair. If you'd like to model with something in between, say 50% of coins are truly fair, and the other 50% is distributed some way among $p\in[0,0.5)\cup(0.5,1]$, then based on that information we could calculate a more interesting answer.
