If matlab shows the number is 1.01010101010101e+41 then you do not have a binary number. You have a decimal integer number consisting of just the digits 0 and 1. Additionally, that decimal number is stored as a 64-bit double.
Any integer number greater than flintmax (= 9007199254740992), so anything with 16 digits or more may not be stored exactly as a double. The greater the number, the less likely you can store it exactly. e.g 100000000000000010 (17 digits) is stored as 100000000000000016. There's nothing that can be done about that, that's the way double numbers work.
You could store your decimal representation as a 64-bit integer instead (uint64) to be able to store a few more digits but that won't get you far. The maximum decimal number consisting of 0s and 1s that you can store as uint64 is 11111111111111111111 (20 digits). Anything above that cannot be stored.
So, the first thing you need to realise is that you cannot store a number consisting of 64 0s and 1s as a decimal number, double or uint64 (unless you defer to the symbolic toolbox but that would be slow).
You could store it as a char vector, as suggested by Bruno.
In my opinion, the simplest way to store a 64-bit number is to store it in its decimal representation as a uint64 (i.e. store the binary 1111111111111111111111111111111111111111111111111111111111111111b as uint64(18446744073709551615)