What has mass but do not take up space? Loosely speaking matter is anything that has mass and takes up space... so photon has no (rest) mass and I don't know if it takes up space but it already been disqualified due to failing to meet criteria number 1. Now I like to know what else isn't matter which do has mass but do not take up space? Which criteria is more crucial in order to qualify as matter? 
 A: You are not specifying it, but you define matter as the constituents of atoms basically, electrons and quarks, that do have rest mass, and are elementary particles. 

In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume.1 All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic particles, and in everyday as well as scientific usage, "matter" generally includes atoms and anything made up of them, and any particles (or combination of particles) that act as if they have both rest mass and volume. However it does not include massless particles such as photons, or other energy phenomena or waves such as light or sound.1[2]

https://en.wikipedia.org/wiki/Matter
Now you are saying that photons are not matter, because they are not constituents of atoms, and are massless.
And you are asking for an elementary particle (as you say does not have volume, meaning pointlike), that does have rest mass. And this particle should not be considered matter (in your case not be constituent of atoms). And I guess you want to exclude antimatter too. It would be very interesting what you would say about the W and Z bosons, having rest mass and being pointlike. But I guess you would say these would be considered having something to do with your definition of matter, contributing to the weak force, which is an elemental necessity for example transformation of matter (proton and neutron). And that I guess excludes the neutrinos too. And the muon and tau are excluded too.

I would give you as an example based on what your definition of matter is, the Higgs boson. The Higgs boson does have rest mass, and is elementary, pointlike, based on your definition, is not a constituent of matter. The Higgs mechanism is what really contributes to the properties (rest mass) of matter, but the Higgs boson (the excitation of the field) itself could be considered based on your definition as not being a constituent of matter.
