Gasoline can and does go stale, it does not last indefinitely. All gasoline sold in Europe has oxydation stability as part of it's specification. This gives you a reasonable guarantee of it's stability, and so should last at least 4 weeks with no problems.
However, there are several factors you should be aware of;
1. High temperatures - will accelerate oxydation and degradation
2. Strong sunlight - the same
3. Bear in mind that your gasoline may not have come straight from the refinery. Traders often store gasoline for many months in tank farms to take advantage of market conditions (higher forward prices), so the gasoline that you receive may already be 6 months old. It will have been tested when it comes out of the tank, but if it fails the oxydation stability test, it will simply have more additive applied to it. The efficacity of such additives can decline with frequent application.
4. The more the gasoline has been 'cracked' at the refinery, the more it tends to be unstable. Therefore, gasoline from simpler refineries will be more stable than that from more complex ones.
5. Some cargoes of gasoline only just meet the oxy stability test, others are well within.
What happens when it oxidises and why is it a problem? Upoin oxydation, deposits can form, and gums and varnishs can appear in the tank, in the lines and in the carbs/injectors. These will obviously cause issues with engine running.
Personally, I now stabilise all my gasoline for safety's sake (for garden machinery and boats). I have had some problems in the past which I put down to old gasoline.
Also, a separate issue is the Reid Vapour Pressure of the gasoline, again part of the specification. This is a measure of how volatile the gasoline is. It is higher in winter than in summer. Over time, volatility will decrease (the hiss you hear when opening a can is the lighter, more volatile fractions escaping). If you have old summer gasoline, that you try to use in winter you may have problems.