You can use 6 month old gas in your mower as long as it smells like good gasoline and is not turning to varnish.
If you've added a fuel stabilizer to the gasoline then it may be still okay at 6 months to use in the mower.
However I would not recommend using 6 month old gas in your mower as it could cause the fuel filter if you have one to clog and even cause the engine to run poorly and gum up the carburetor.
I would recommend getting some fresh gasoline to use in the lawnmower instead.
Using gasoline that is more than 30 days old in a lawn mower is not recommended as it can degrade over time and lose it's combustibility and effectiveness and not ignite properly so the lawn mower engine may be hard to start, stall or not run at all.
Gas when stored in a gas can is often good for around 2 to 3 months.
In some cases the gas in the gas can may last up to 6 months before it goes bad.
Gasoline that has a higher ethanol content may degrade and go bad faster than gasoline without ethanol.
Adding a fuel stabilizer to the gasoline when storing it can also help extend the life of the gasoline
However it's always best to use the gasoline in the can within a month or less of buying it and then add fresh gas to the gas can to keep fresh gasoline available.
Always keep the gasoline in a can stored in a ventilated garage or shed that is not attached to your home.
And avoid leaving the gasoline can and gasoline out and exposed to heat, sunlight and moisture which also shortens the lifespan of the gasoline.
If the gasoline is starting to smell like varnish and is starting to turn to varnish then it's going bad and in that case it's not recommended to use it as it could damage fuel injectors, clog fuel filters, carburetors etc.