How can people save in a equilibrium economy? Two people on a desert island:


John produces 100kg potatoes and sells them to Paul for \$100. Paul catches 100kg fish and sells it to John for \$100. GDP is \$200\$, right?
Next year, they improve their techniques and produce 150kg of potatoes and fish, respectively. 
Now, John would love to sell 150kg potatoes to Paul for \$150 and buy only 100kg of fish for \$100, so he can save apart \$50. Unfortunately, Paul has the same wish!
It seems there is no way they can both save. If both transact \$100, the GDP will be like last year. If both transact \$150 the GDP will be \$300. 
In both cases, the economy is at equilibrium. It can grow or stay the same. But no saving is possible!
It seems that, in order for John to save \$50, Paul must consume more than his income (or the other way around).


How then is saving in the real economy possible? 

Is it a zero sum game, that is, for every John saving, there must be a Paul that consumes more than he can afford?
The question is belied by some basic misconceptions. Just to list a few:


The comparison of GDP in nominal terms and implied statements about growth.
The definition/measurement of GDP (where saving is apparently not part of GDP).
The (completely) arbitrary prices of goods being prescribed for this economy.
The meaning of equilibrium. 



  John produces 100kg potatoes...Paul
  catches 100kg fish...
  Next year, they improve their techniques and produce 150kg of potatoes
  and fish, respectively.


In this example, there is real GDP growth, due to technological innovation.
There are more goods in the economy---more potatoes and more fish. Therefore GDP increases.
This has nothing to do with trade between John and Paul or whether money (some medium of exchange called $---e.g. rocks) is being used to intermediate trade in this economy.

Macroeconomic growth is measured in real terms, not nominal terms.  

As already pointed out by other answers, saving in the macroeconomy is simply output that is not consumed. If John and Paul barter (no money) and at the end consume 100kg fish and 80kg potatoes between the two of them, then aggregate saving is 20kg potatoes. That 20kg potatoes is still part of GDP. 

(The money you put in your savings account is part of your country's GDP.)

Next, consider the proposed scenario:


  Now, John would love to sell 150kg potatoes to Paul for \$150 and buy
  only 100kg of fish for \$100, so he can save apart \$50. Unfortunately,
  Paul has the same wish!


Ok, suppose money, say rocks, is being used in this economy.
(The answer by @1muflon1 touches upon monetary policy by a monetary authority---e.g. there is a central bank on this island that sets the interest rate.)

Where do these prices ( \$1/kg for potatoes and fish) and the demand for saving \$50 you assume come from?

Suppose for the moment there's no saving technology---e.g. John cannot put his money under his bed.
If price of potatoes and fish are $\$P_p$ and $\$P_f$ per kg.
Then John has $ \$150 P_p $ and based on this budget he would choose some quantity
$Q_p$ and $Q_f$ of potatoes and fish so that his budget balances:
$$
Q_p P_p + Q_f P_f = 150 P_p.
$$
Same goes for Paul. 

The prices $P_p$ and $P_f$ that actually realize in the economy are such that the potato and fish markets clear, i.e. total amount of potatoes demanded equal to total supply 100 kg, same for fish.

If at price $\$ P_p'$ per kg, John and Paul are willing to purchase only 90 kg of potatoes between them, then $\$ P_p'$ is too high a price, and in equilibrium you would never see potatoes priced at $\$ P_p'$ per kg.

Now suppose there is a saving technology---e.g. John can put money under his bed to spend next year.

Saving means shifting consumption from today to tomorrow. The amount John chooses to save depends on what \$1 represents in real terms tomorrow, which in turn is determined by price. John wants to save \$X today because \$X will buy him certain amount of potatoes and fish tomorrow.

If, at the prevailing price \$1/kg for potato and fish, John would choose to save 
\$50, this means John would optimally like to have 50kg of, say, potatoes tomorrow.

However, if there are excess supply of potatoes and fish at the prevailing price of \$1/kg (as in the proposed scenario), prices will decrease, in order to clear the market. As today's potato and fish prices decrease, it becomes more attractive for John to consume today rather than save.
Or the lower prices today leads him to expect lower prices tomorrow and thus he would need to save less \$. His optimal saving choice would change accordingly.

Therefore the above scenario would not occur in this economy:


  Now, John would love to sell 150kg potatoes to Paul for \$150 and buy
  only 100kg of fish for \$100, so he can save apart $50. Unfortunately,
  Paul has the same wish!


If saving $50 by John is infeasible in this economy, then it cannot be an optimal saving choice for John.
In equilibrium, prices equilibriate so that optimal choices of economic agents are jointly feasible.


  In both cases, the economy is at equilibrium.


No, what "equilibrium"?


  How then is saving in the real economy possible?


It should be clear that this question is based on several strands of incorrect reasoning.


  Is it a zero sum game, that is, for every John saving, there must be a
  Paul that consumes more than he can afford?


As pointed out by @1muflon1, there are certain macroeconomic scenarios where saving becomes something like a zero-sum game. Not in the sense you describe, but in the sense that saving by one agent translates to loss of potential wealth for another agent.

Suppose there's a central bank on the island that sets interest rate on saving. The raising and lowering of interest rate would influence John's (and Paul's) consumption/saving decision. Lowering interest rate induces John to consume more today, which increases his demand for fish, which increases price of fish, therefore more wealth for Paul. Vice versa. This is a simplistic version of the central bank lowering rates to stimulate the economy/generate inflation.

However, if rates are sufficiently low ("zero lower bound", as you can find in macro texts), John and Paul might prefer to hoard cash, i.e. liquidity, rather than spend.
This is the Keynesian liquidity trap where monetary policy becomes ineffective and there is excessive saving.

(The liquidity-hoarding instead of spending on fish/potatoes story is awkward for two people on an island. In the actual macroeconomy, people hoard liquidity because return on bonds, which are less liquid than cash, is less than the liquidity premium.)