The floating point arithmetic format that occupies 128 bits of storage is known as binary128 or quadruple precision.
The following blog post by MathWorks' Chief Mathematician Cleve Moler describes an implementation of quadruple precision programmed entirely in the MATLAB language:
Please note that quadruple precision is not available as of MATLAB R2018b. As a workaround, it is possible to obtain more precision using Variable Precision Arithmetic from the Symbolic Math Toolbox. Another possibility is to use the Fixed-Point Toolbox to obtain more precision. Note that in either case, the numbers will not be of the floating point format.