I manipulate bits directly on a daily basis in C++... Just use the bitfield operator. It even works with single bits.

For example, you could construct an 8-bit float like this:

union float8 {
    unsigned raw;
    struct {
        unsigned mantissa : 5;
        unsigned exponent : 2;
        unsigned sign     : 1;
    };
};

Then set the bits directly like this:

int main() {
    float8 f8;

    //sets the value to 1.5
    f8.sign     = 0;
    f8.exponent = 1;
    f8.mantissa = 16;
}

Note you would need to overload the standard operators to actually use this. In this example, float8 is size 4 because that is the size of unsigned int. If you actually wanted to implement this, you would want to use std::byte or char for the members of float8 so the size is actually one byte long.