.333... Is a third. That's just a quirk of base 10. If you go to a different number system you won't run into that particular issue.
The most common other base people know of is binary. Base 2. So in binary the fraction would be 1/11 and then 1/11(binary)=1/3(base 10).
I remember talk back in the day that base 12 is good for most common human problems. Some people were interested in trying to get people to switch to that.
1/3 of 12 is 4.
So 4/12=1/3=3.33333.../10
.333... Is just the cursive way of writing 1/3.
I still don't "grasp" infinity. I'd recon you'd need an infinite mind to grasp infinity.