Sure, it's possible, but to rephrase my question: why isn't it standard already? I mean, the last C standard was published in 2017, long after Big Integers became a need. And still, the largest standard integer you can define is 64-bits long.
I suppose it is because of hardware limitations. We haven't really improved a lot of things in CPUs, x64 was released 30 years ago and there is no x128 on the horizons or for example core speed (clock rate) hasn't really improved for decades (we just get more cores to have faster CPU).
Our hardware simply can not handle integers bigger than 64-bit (no bigger registers). So it has to be implemented using smaller chunks which is what loads of existing math/arithmetic libraries do so I don't think the standard itself needs to have them.
However, many languages have added the bigger integer types. You already know the one in C++, there is also
Int128 in dotnet. It took them 7 version of core and at least 5 years to implement it and as you can see it doesn't do anything special, it uses 2 UInt64 limbs and the arithmetic is
a bunch of branches handling the overflow which is not the fastest thing.